{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":43755,"title":"Divide the Least Common Multiple by the Greatest Common Divisor of two numbers","description":"Divide the Least Common Multiple by the Greatest Common Divisor of two numbers. For example, for x=12345 and y=54321, the answer would be\r\n\r\n 223530915/3 =\r\n 74510305 (ANSWER)","description_html":"\u003cp\u003eDivide the Least Common Multiple by the Greatest Common Divisor of two numbers. For example, for x=12345 and y=54321, the answer would be\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003e223530915/3 =\r\n74510305 (ANSWER)\r\n\u003c/pre\u003e","function_template":"function z = your_fcn_name(x,y)\r\n y = x;\r\nend","test_suite":"%%\r\nx = 12345;\r\ny = 54321;\r\ny_correct = 74510305;\r\nassert(isequal(your_fcn_name(x,y),y_correct))\r\n%%\r\nx = 12;\r\ny = 54;\r\ny_correct = 18;\r\nassert(isequal(your_fcn_name(x,y),y_correct))\r\n%%\r\nx = 1;\r\ny = 1;\r\ny_correct = 1;\r\nassert(isequal(your_fcn_name(x,y),y_correct))\r\n%%\r\nx = 987654321;\r\ny = x;\r\ny_correct = 1;\r\nassert(isequal(your_fcn_name(x,y),y_correct))\r\n%%\r\nx = 800;\r\ny = 26000;\r\ny_correct = 130;\r\nassert(isequal(your_fcn_name(x,y),y_correct))\r\n%%\r\nx = 65536;\r\ny = 32768;\r\ny_correct = 2;\r\nassert(isequal(your_fcn_name(x,y),y_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":93456,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":53,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2016-12-07T23:00:01.000Z","updated_at":"2026-03-19T18:26:49.000Z","published_at":"2016-12-07T23:00:01.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDivide the Least Common Multiple by the Greatest Common Divisor of two numbers. For example, for x=12345 and y=54321, the answer would be\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[223530915/3 =\\n74510305 (ANSWER)]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":49327,"title":"Divisors","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 21px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 343px 10.5px; transform-origin: 343px 10.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 320px 10.5px; text-align: left; transform-origin: 320px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eFind the divisors of a given number. But don't return the number itself, neither 1.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = divisores(n)\r\n y = n;\r\nend","test_suite":"%%\r\nx = 20;\r\ny_correct = [2,4,5,10];\r\nassert(isequal(divisores(x),y_correct))\r\n%%\r\nx = 30;\r\ny_correct = [2,3,5,6,10,15];\r\nassert(isequal(divisores(x),y_correct))\r\n%%\r\nx = 77;\r\ny_correct = [7,11];\r\nassert(isequal(divisores(x),y_correct))\r\n%%\r\nx = 101; %dalmatians\r\ny_correct = [];\r\nassert(isequal(divisores(x),y_correct))\r\n%%\r\nx = 777;\r\ny_correct = [3,7,21,37,111,259];\r\nassert(isequal(divisores(x),y_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":698530,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":40,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-12-23T11:05:39.000Z","updated_at":"2026-02-28T08:27:11.000Z","published_at":"2020-12-31T01:21:53.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFind the divisors of a given number. But don't return the number itself, neither 1.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":44748,"title":"Amicable numbers","description":"Test whether two numbers are \u003chttps://en.wikipedia.org/wiki/Amicable_numbers amicable\u003e, meaning that the sum of the proper divisors of each number is equal to the other number.\r\n\r\n\r\nExample: 220 and 284 are amicable numbers because the proper divisors of 220 are 1,2,4,5,10,11,20,22,44,55,110 and their sum is 284, while the proper divisors of 284 are 1,2,4,71,142 and their sum is 220.","description_html":"\u003cp\u003eTest whether two numbers are \u003ca href = \"https://en.wikipedia.org/wiki/Amicable_numbers\"\u003eamicable\u003c/a\u003e, meaning that the sum of the proper divisors of each number is equal to the other number.\u003c/p\u003e\u003cp\u003eExample: 220 and 284 are amicable numbers because the proper divisors of 220 are 1,2,4,5,10,11,20,22,44,55,110 and their sum is 284, while the proper divisors of 284 are 1,2,4,71,142 and their sum is 220.\u003c/p\u003e","function_template":"function y = amicable(m,n)\r\n y = false;\r\nend","test_suite":"%%\r\nm = 220; n = 284;\r\ny_correct = true;\r\nassert(isequal(amicable(m,n),y_correct))\r\n\r\n%%\r\nm = 220; n = 504;\r\ny_correct = false;\r\nassert(isequal(amicable(m,n),y_correct))\r\n\r\n%%\r\nm = 2620; n = 2924;\r\ny_correct = true;\r\nassert(isequal(amicable(m,n),y_correct))\r\n\r\n%%\r\nm = 5020; n = 5564;\r\ny_correct = true;\r\nassert(isequal(amicable(m,n),y_correct))\r\n\r\n%%\r\nm = 2924; n = 5020;\r\ny_correct = false;\r\nassert(isequal(amicable(m,n),y_correct))","published":true,"deleted":false,"likes_count":3,"comments_count":4,"created_by":254267,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":78,"test_suite_updated_at":"2018-10-22T18:10:27.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2018-10-22T17:57:52.000Z","updated_at":"2026-03-16T15:34:17.000Z","published_at":"2018-10-22T18:02:09.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTest whether two numbers are\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Amicable_numbers\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eamicable\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, meaning that the sum of the proper divisors of each number is equal to the other number.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample: 220 and 284 are amicable numbers because the proper divisors of 220 are 1,2,4,5,10,11,20,22,44,55,110 and their sum is 284, while the proper divisors of 284 are 1,2,4,71,142 and their sum is 220.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":45794,"title":"*Prime number check 2 (in construction)","description":"Another way to see if a number is prime is to count the number of factors. For example,\r\n\r\n the number 4 has 2 factors, [ 2 4 ]\r\n the number 16 has 4 factors, [ 2 4 8 16 ]\r\n the number 7 has 1 factor, [ 7 ]\r\n\r\nThus, if a number has only one factor, it is prime. We ignore the factor 1 as it is trivial. \r\n\r\nYour function should create a vector containing all the factors of any number x.","description_html":"\u003cp\u003eAnother way to see if a number is prime is to count the number of factors. For example,\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003ethe number 4 has 2 factors, [ 2 4 ]\r\nthe number 16 has 4 factors, [ 2 4 8 16 ]\r\nthe number 7 has 1 factor, [ 7 ]\r\n\u003c/pre\u003e\u003cp\u003eThus, if a number has only one factor, it is prime. We ignore the factor 1 as it is trivial.\u003c/p\u003e\u003cp\u003eYour function should create a vector containing all the factors of any number x.\u003c/p\u003e","function_template":"function y = your_fcn_name(x)\r\n y = x;\r\nend","test_suite":"%%\r\nx = 16;\r\ny_correct = [ 2 4 8 16 ];\r\nassert(isequal(your_fcn_name(x),y_correct))\r\n%%\r\nx = 23;\r\ny_correct = [ 23 ];\r\nassert(isequal(your_fcn_name(x),y_correct))\r\n%%\r\nx = 100;\r\ny_correct = [ 2 4 5 10 20 25 50 100 ];\r\nassert(isequal(your_fcn_name(x),y_correct))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":428668,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":29,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-06-07T20:00:06.000Z","updated_at":"2025-08-03T17:44:27.000Z","published_at":"2020-06-07T20:00:06.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAnother way to see if a number is prime is to count the number of factors. For example,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[the number 4 has 2 factors, [ 2 4 ]\\nthe number 16 has 4 factors, [ 2 4 8 16 ]\\nthe number 7 has 1 factor, [ 7 ]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThus, if a number has only one factor, it is prime. We ignore the factor 1 as it is trivial.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYour function should create a vector containing all the factors of any number x.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":1933,"title":"That's some divisor you've got there...","description":"Given a positive integer x, calculate the sum of all of the divisors of the number. Please include the number itself in your final answer. For example, 1, 3, 5 and 15 are all the proper divisors of 15, so sum_divisors(15)=1+3+5+15, which is 24.\r\n\r\nYour score will be based on how quickly your script solves the problems in the test suite, rather than the typical Cody size. Good luck!","description_html":"\u003cp\u003eGiven a positive integer x, calculate the sum of all of the divisors of the number. Please include the number itself in your final answer. For example, 1, 3, 5 and 15 are all the proper divisors of 15, so sum_divisors(15)=1+3+5+15, which is 24.\u003c/p\u003e\u003cp\u003eYour score will be based on how quickly your script solves the problems in the test suite, rather than the typical Cody size. Good luck!\u003c/p\u003e","function_template":"function y = sum_divisors(x)\r\n y = x;\r\nend","test_suite":"%%\r\nt_in=clock;\r\n\r\nx = 1; y_correct = 1; yours=sum_divisors(x);\r\nassert(isequal(yours,y_correct))\r\nt_out=etime(clock,t_in)*1000;\r\nfprintf('Actual Time = %.0f msec\\n',t_out)\r\n\r\nx = 2; y_correct = 3; yours=sum_divisors(x);\r\nassert(isequal(yours,y_correct))\r\nt_out=etime(clock,t_in)*1000;\r\nfprintf('Actual Time = %.0f msec\\n',t_out)\r\n\r\nx = 120; y_correct = 360; yours=sum_divisors(x);\r\nassert(isequal(yours,y_correct))\r\nt_out=etime(clock,t_in)*1000;\r\nfprintf('Actual Time = %.0f msec\\n',t_out)\r\n\r\n% Perfect Number!\r\nx = 33550336; y_correct = 67100672; yours=sum_divisors(x);\r\nassert(isequal(yours,y_correct))\r\nt_out=etime(clock,t_in)*1000;\r\nfprintf('Actual Time = %.0f msec\\n',t_out)\r\n\r\nx = 223092870; y_correct = 836075520; yours=sum_divisors(x);\r\nassert(isequal(yours,y_correct))\r\nt_out=etime(clock,t_in)*1000;\r\nfprintf('Actual Time = %.0f msec\\n',t_out)\r\n\r\nx = 4294967295; y_correct = 7304603328; yours=sum_divisors(x);\r\nassert(isequal(yours,y_correct))\r\nt_out=etime(clock,t_in)*1000;\r\nfprintf('Actual Time = %.0f msec\\n',t_out)\r\n\r\nx=arrayfun(@(y) sum_divisors(y),1:20000);assert(isequal(sum(x),329004151));\r\nt_out=etime(clock,t_in)*1000;\r\nassert(all(x\u003e0));\r\nfprintf('Actual Time = %.0f msec\\n',t_out)\r\n\r\nx=arrayfun(@(p) sum_divisors(p),primes(200000));assert(isequal(sum(x),1709618797));\r\nt_out=etime(clock,t_in)*1000;\r\nassert(all(x\u003e0));\r\n\r\nudx=unique(diff(x));\r\nassert(isequal(numel(udx),39))\r\nassert(isequal(sum_divisors(max(udx)),132))\r\n\r\nt2=min(100000,t_out);\r\nfprintf('Actual Time = %.0f msec\\n',t_out)\r\nfeval(@assignin,'caller','score',floor(t2));","published":true,"deleted":false,"likes_count":4,"comments_count":0,"created_by":1615,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":36,"test_suite_updated_at":"2018-08-22T14:05:40.000Z","rescore_all_solutions":false,"group_id":38,"created_at":"2013-10-11T15:21:25.000Z","updated_at":"2025-03-08T05:05:07.000Z","published_at":"2013-10-11T15:21:25.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a positive integer x, calculate the sum of all of the divisors of the number. Please include the number itself in your final answer. For example, 1, 3, 5 and 15 are all the proper divisors of 15, so sum_divisors(15)=1+3+5+15, which is 24.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYour score will be based on how quickly your script solves the problems in the test suite, rather than the typical Cody size. Good luck!\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":60935,"title":"Compute the antisigma function","description":"The sum of divisors function, or , is the sum of the numbers that divide . The antisigma function is the sum of numbers less than that do not divide . For example, antisigma(15) = 2+4+6+7+8+9+10+11+12+13+14 = 96. \r\nWrite a function that computes the antisigma function. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 72px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 36px; transform-origin: 408px 36px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 99.175px 8px; transform-origin: 99.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe sum of divisors function, or \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"30.5\" height=\"18.5\" alt=\"sigma(n)\" style=\"width: 30.5px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 120.175px 8px; transform-origin: 120.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, is the sum of the numbers that divide \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 145.208px 8px; transform-origin: 145.208px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The antisigma function is the sum of numbers less than \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 57.175px 8px; transform-origin: 57.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that do not divide \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 219.358px 8px; transform-origin: 219.358px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. For example, antisigma(15) = 2+4+6+7+8+9+10+11+12+13+14 = 96. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 168.283px 8px; transform-origin: 168.283px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that computes the antisigma function. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = antisigma(n)\r\n y = n-sum(n);\r\nend","test_suite":"%%\r\nassert(isequal(antisigma(0),0))\r\n\r\n%%\r\nassert(isequal(antisigma(1),0))\r\n\r\n%%\r\nassert(isequal(antisigma(15),96))\r\n\r\n%%\r\nassert(isequal(antisigma(66),2067))\r\n\r\n%%\r\nassert(isequal(antisigma(214),22681))\r\n\r\n%%\r\nassert(isequal(antisigma(651),211202))\r\n\r\n%%\r\nassert(isequal(antisigma(2030),2057145))\r\n\r\n%%\r\nassert(isequal(antisigma(4306),9266509))\r\n\r\n%%\r\nassert(isequal(antisigma(6771),22917182))\r\n\r\n%%\r\nassert(isequal(antisigma(8888),39484356))\r\n\r\n%%\r\nassert(isequal(antisigma(9999),49979088))\r\n\r\n%%\r\nassert(isequal(antisigma(45364),1028882242))\r\n\r\n%%\r\nassert(isequal(antisigma(351433),61752399801))\r\n\r\n%%\r\nassert(isequal(antisigma(7665534),29380192820025))\r\n\r\n%%\r\nassert(isequal(antisigma(88432534),3910156446398041))\r\n\r\n%%\r\np = primes(1e5);\r\nq = p(randi(length(p)));\r\nassert(isequal(antisigma(q),polyval([1/2 -1/2 -1],q)))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2025-06-12T01:47:32.000Z","deleted_by":null,"deleted_at":null,"solvers_count":8,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-06-12T01:47:03.000Z","updated_at":"2025-10-14T02:58:47.000Z","published_at":"2025-06-12T01:47:32.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe sum of divisors function, or \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"sigma(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, is the sum of the numbers that divide \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The antisigma function is the sum of numbers less than \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e that do not divide \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. For example, antisigma(15) = 2+4+6+7+8+9+10+11+12+13+14 = 96. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that computes the antisigma function. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":42791,"title":"Number of divisors of a given number","description":"Given a Number n, return the number of his divisors without listing them\r\n\r\nexample: \r\n\r\nn=14\r\n; Divisors={1,7,2,14} ; y=4\r\n\r\nn=68\r\n; Divisors={1,2,34,17,4,68} ; y=6","description_html":"\u003cp\u003eGiven a Number n, return the number of his divisors without listing them\u003c/p\u003e\u003cp\u003eexample:\u003c/p\u003e\u003cp\u003en=14\r\n; Divisors={1,7,2,14} ; y=4\u003c/p\u003e\u003cp\u003en=68\r\n; Divisors={1,2,34,17,4,68} ; y=6\u003c/p\u003e","function_template":"function y = your_fcn_name(n)\r\n\r\nend","test_suite":"%%\r\nfiletext = fileread('your_fcn_name.m');\r\nassert(isempty(strfind(filetext, 'sqrt')))\r\nassert(isempty(strfind(filetext, 'for')))\r\n\r\n%%\r\nn= 6880;\r\ny_correct = 24;\r\nassert(isequal(your_fcn_name(n),y_correct))\r\n\r\n%%\r\nn= 5050;\r\ny_correct = 12;\r\nassert(isequal(your_fcn_name(n),y_correct))\r\n\r\n%%\r\nn= 188325;\r\ny_correct = 36;\r\nassert(isequal(your_fcn_name(n),y_correct))\r\n\r\n\r\n%%\r\nn= 76576500;\r\ny_correct = 576;\r\nassert(isequal(your_fcn_name(n),y_correct))\r\n\r\n%%\r\nn= 74;\r\ny_correct = 4;\r\nassert(isequal(your_fcn_name(n),y_correct))\r\n\r\n\r\n%%\r\nn=14^8 ;\r\ny_correct = 81;\r\nassert(isequal(your_fcn_name(n),y_correct))\r\n\r\n\r\n%%\r\nn=314721 ;\r\ny_correct = 27;\r\nassert(isequal(your_fcn_name(n),y_correct))","published":true,"deleted":false,"likes_count":2,"comments_count":6,"created_by":17228,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":66,"test_suite_updated_at":"2016-04-01T06:25:39.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2016-03-31T12:09:31.000Z","updated_at":"2025-12-03T20:02:50.000Z","published_at":"2016-03-31T12:09:31.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a Number n, return the number of his divisors without listing them\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eexample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003en=14 ; Divisors={1,7,2,14} ; y=4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003en=68 ; Divisors={1,2,34,17,4,68} ; y=6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":59841,"title":"Count the ways to write 1/n as the sum of two unit fractions","description":"The number 1/2 can be written as 1/3+1/6 and 1/4+1/4, and the number 1/9 can be written as 1/18+1/18, 1/12+1/36, and 1/10+1/90. That is, 1/2 can be written as the sum of two unit fractions (those with a numerator of 1) in two ways, and 1/9 can be written as the sum of two unit fractions in three ways. \r\nWrite a function to count the ways can be written as the sum of two unit fractions.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 93px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 46.5px; transform-origin: 407px 46.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 374.208px 8px; transform-origin: 374.208px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe number 1/2 can be written as 1/3+1/6 and 1/4+1/4, and the number 1/9 can be written as 1/18+1/18, 1/12+1/36, and 1/10+1/90. That is, 1/2 can be written as the sum of two unit fractions (those with a numerator of 1) in two ways, and 1/9 can be written as the sum of two unit fractions in three ways. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 106.825px 8px; transform-origin: 106.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to count the ways \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"25.5\" height=\"18.5\" alt=\"1/n\" style=\"width: 25.5px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 145.842px 8px; transform-origin: 145.842px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e can be written as the sum of two unit fractions.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = twoUnitFractions(n)\r\n y = 1/n+1/n;\r\nend","test_suite":"%%\r\nn = 2;\r\ny = twoUnitFractions(n);\r\ny_correct = 2;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 9;\r\ny = twoUnitFractions(n);\r\ny_correct = 3;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 85;\r\ny = twoUnitFractions(n);\r\ny_correct = 5;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 327;\r\ny = twoUnitFractions(n);\r\ny_correct = 5;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 461;\r\ny = twoUnitFractions(n);\r\ny_correct = 2;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 824;\r\ny = twoUnitFractions(n);\r\ny_correct = 11;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 1989;\r\ny = twoUnitFractions(n);\r\ny_correct = 23;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 5460;\r\ny = twoUnitFractions(n);\r\ny_correct = 203;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 7011;\r\ny = twoUnitFractions(n);\r\ny_correct = 23;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 8610;\r\ny = twoUnitFractions(n);\r\ny_correct = 122;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 9438;\r\ny = twoUnitFractions(n);\r\ny_correct = 68;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 14332;\r\ny = twoUnitFractions(n);\r\ny_correct = 8;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 58110;\r\ny = twoUnitFractions(n);\r\ny_correct = 122;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 254000;\r\ny = twoUnitFractions(n);\r\ny_correct = 95;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 5563449;\r\ny = twoUnitFractions(n);\r\ny_correct = 8;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 77221133;\r\ny = twoUnitFractions(n);\r\ny_correct = 5;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\np = primes(1000);\r\ny = arrayfun(@twoUnitFractions,p);\r\nassert(all(y==2))\r\n\r\n%%\r\np = primes(20);\r\nb = p(randi(length(p)));\r\ne = randi(5);\r\ny = twoUnitFractions(b^e);\r\nassert(isequal(y,e+1))\r\n\r\n%%\r\nfiletext = fileread('twoUnitFractions.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2024-04-05T01:09:45.000Z","deleted_by":null,"deleted_at":null,"solvers_count":7,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2024-04-05T01:09:23.000Z","updated_at":"2025-07-19T12:22:30.000Z","published_at":"2024-04-05T01:09:45.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe number 1/2 can be written as 1/3+1/6 and 1/4+1/4, and the number 1/9 can be written as 1/18+1/18, 1/12+1/36, and 1/10+1/90. That is, 1/2 can be written as the sum of two unit fractions (those with a numerator of 1) in two ways, and 1/9 can be written as the sum of two unit fractions in three ways. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to count the ways \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"1/n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e1/n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e can be written as the sum of two unit fractions.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":2664,"title":"Divisors for big integer","description":"Inspired by Problem 1025 and Project Euler 12.\r\n\r\nGiven n, return the number y of integers that divide N. \r\n\r\nFor example, with n = 10, the divisors are [1 2 5 10], so y = 4.\r\n\r\nIt's easy with normal integer but how to proceed with big number?\r\n\r\n","description_html":"\u003cp\u003eInspired by Problem 1025 and Project Euler 12.\u003c/p\u003e\u003cp\u003eGiven n, return the number y of integers that divide N.\u003c/p\u003e\u003cp\u003eFor example, with n = 10, the divisors are [1 2 5 10], so y = 4.\u003c/p\u003e\u003cp\u003eIt's easy with normal integer but how to proceed with big number?\u003c/p\u003e","function_template":"function y = divisors_Big(x)\r\n y = x;\r\nend","test_suite":"%%\r\nx = 10;\r\ny_correct = 4;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 28;\r\ny_correct = 6;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 28;\r\ny_correct = 6;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 784;\r\ny_correct = 15;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 1452637;\r\ny_correct = 2;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 5452637;\r\ny_correct = 4;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 16452637;\r\ny_correct = 2;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 116452637;\r\ny_correct = 8;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 416452638;\r\ny_correct = 32;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 12250000;\r\ny_correct = 105;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 2031120;\r\ny_correct = 240;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 76576500;\r\ny_correct = 576;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 816452637;\r\ny_correct = 32;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 103672800;\r\ny_correct = 648;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 842161320;\r\ny_correct = 1024;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n\r\n","published":true,"deleted":false,"likes_count":4,"comments_count":0,"created_by":5390,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":240,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2014-11-12T02:00:29.000Z","updated_at":"2026-01-05T00:22:49.000Z","published_at":"2014-11-12T02:28:56.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eInspired by Problem 1025 and Project Euler 12.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven n, return the number y of integers that divide N.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, with n = 10, the divisors are [1 2 5 10], so y = 4.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIt's easy with normal integer but how to proceed with big number?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":46898,"title":"Sum of all the divisors of n","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 21px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 10.5px; transform-origin: 407px 10.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eOutput the sum of all the divisors of a number (n). For example n=10, divisors = 1, 2, 5, 10, output = 18.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = sumDivisors(x)\r\n y = sum(x);\r\nend","test_suite":"%%\r\nx=2.^(10:20);\r\ny=[2047,4095,8191,16383,32767,65535,131071,262143,524287,1048575,2097151];\r\nassert(isequal(sumDivisors(x),y))\r\n%%\r\nx=2^32-1;\r\ny=7304603328;\r\nassert(isequal(sumDivisors(x),y))\r\n%%\r\nx =[ 53394 10330 42975 46272 28754 18090 49236 55098 23047 4972\r\n 59362 63609 2341 2087 25006 44545 16718 16665 54450 3536\r\n 8323 62729 55649 18149 50169 42933 33159 53365 38356 34787\r\n 59860 31810 61211 3026 52115 10657 45815 15960 36027 51064\r\n 41443 52448 44482 6366 12247 7799 58387 60901 60110 61212\r\n 6393 9299 49660 53967 32098 32661 62869 22937 18733 8514\r\n 18252 27641 48702 45537 29202 62898 35863 12885 49624 37279\r\n 35841 60014 25705 20782 42357 22308 9085 16456 49397 30762\r\n 62752 51919 42958 62274 46489 38357 9785 40374 24933 781\r\n 63235 62882 11219 2258 49460 14668 16877 31018 37213 22094];\r\ny=[ 116640 18612 77376 122936 47088 48960 125664 119418 24280 9576\r\n 90576 104832 2342 2088 37512 54720 27048 29376 160797 7812\r\n 10080 63300 60720 18150 76480 62496 50560 69048 68992 35640\r\n 130536 57276 61212 4860 71520 10658 73872 57600 52052 103320\r\n 41444 113400 69696 12744 12616 8520 70400 60902 108216 142856\r\n 8528 9864 112896 71960 52560 49920 62870 22938 22176 20592\r\n 51240 27984 97416 62304 60672 137376 35864 20640 93060 40680\r\n 51520 92568 31752 31176 64576 61488 11520 35910 50496 66690\r\n 129276 59344 65952 127008 46490 42294 12480 87516 33248 864\r\n 75888 98496 12096 3390 103908 27160 19296 50148 43200 33144];\r\nassert(isequal(sumDivisors(x),y))\r\n\r\n","published":true,"deleted":false,"likes_count":4,"comments_count":0,"created_by":145982,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":26,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-10-18T03:07:30.000Z","updated_at":"2025-11-29T15:06:14.000Z","published_at":"2020-10-18T03:27:54.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOutput the sum of all the divisors of a number (n). For example n=10, divisors = 1, 2, 5, 10, output = 18.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51715,"title":"Iterate the sum of divisors and totient","description":"","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 339px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 169.5px; transform-origin: 407px 169.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46898\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003eCody Problem 46898\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 160.25px 7.91667px; transform-origin: 160.25px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e deals with the sum of divisors function, denoted by \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAD0AAAAmCAYAAAB3c5OxAAACSElEQVRoge1YW5GEMBBsDzjAAAZQsArWAQ5wsBbQgIT1gAU0YGHvI+ljNpeQyV5CcVfpKr4Set4zSYCKioqKimQ8AHSZuFrL12TiK4IJwD0zZw9gwUUNnwCMhbjvMIZfCgPKKzXBpPol0AJ4waRhSTQAthPkqDABeJ4k6wFgPUlWEIzy7Z/K8+JhlTgTC4A5F1kL48Hx4HPn74q01O6sDBmpBqY7j9BFMIujexjFX4pPzkqmWmxMDR5+zvIbTHOSazHD7/hl45wcgatVYrOKym9y/u3tP4NS1mL3b9iju1qem9AhNpZS5b5hFoI27B7Wko5I8/hq98/2nxV75lCmhq+DLsN+gHXB6LpHPGbAkeEpRrMUyMkIu1zMghjoPDWkV0PDXqZbyKgUo1mHdLLrTGad1pAXEs8Gso5D9dMr9qQYLUvJp6zMAg2SjZaNqw3skZEO1U6K0ezQG36OPengkD4S2qnxDTaBmKdkNoTuxnRMbMRIo9wJAOz9RXu8JJ/aaE3a8mAfUpLQdlHZNH0O5CijPrFo09nqOS27aMhoRtnX1V1siDcfGuWLpNSnt/KeOH55YVklPSrwdOS7/w5iTUM64TgtGxz3hgHv9bwg/vKy4IO7e4P3lOKJ6An/ODkCUy0UGTmqfOko1zXdm078+EnKvVx8+pi3Ilz7veAPYbDrms49QH+AKQpeGs5QJDUTi2LGcafPgREXexxknyj1otHBZJOmBE4Fu2+uh36CDs3Nmw2aOZvKN2fkK4pcTa3JyFVRUVHxd/AF1+TpTklxPxAAAAAASUVORK5CYII=\" alt=\"sigma(n)\" style=\"width: 30.5px; height: 19px;\" width=\"30.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.7833px 7.91667px; transform-origin: 21.7833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, while \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/656\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003eCody Problem 656\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 46.675px 7.91667px; transform-origin: 46.675px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e deals with the totient function, denoted by \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"phi(n)\" style=\"width: 31.5px; height: 19px;\" width=\"31.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 164.125px 7.91667px; transform-origin: 164.125px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The sum of divisors is straightforward: for example, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"sigma(12) = 1+2+3+4+6+12 = 28\" style=\"width: 227.5px; height: 19px;\" width=\"227.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The totient of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 95.2917px 7.91667px; transform-origin: 95.2917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e counts the numbers less than \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 84.4px 7.91667px; transform-origin: 84.4px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that are relatively prime to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 44.725px 7.91667px; transform-origin: 44.725px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. For example \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"phi(12) = 4\" style=\"width: 65px; height: 19px;\" width=\"65\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68.5083px 7.91667px; transform-origin: 68.5083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e because the greatest common divisor of 12 and four numbers (1, 5, 7, 11) is 1. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 384px 7.91667px; transform-origin: 384px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWhat happens if you repeatedly apply the two functions, starting with the sum of divisors and alternating? For example, start with 7. Then \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAvMAAAAmCAYAAABJaRRiAAAQC0lEQVR4nO2d63HrOAyFTw/uwA24AVfgCtxBOnAHacE1pIT0kBZSQ1q4+0M+K4ThA3xZpIVvRrOzvo4skQRwCFIQYBiGYRiGYRiGYRiGYRiGYRiGYRiGYRiGYRiGYRiGYRiGYRiGYRiGYRiGYRg75x3AqdG5jo/zHRqdz2jDAcAd7frlAuCt0bmMelraMLCMlWPD8xn1tLbhPfrqkcf09XGUcsDvvrwBOFddkTEarf18DRYjBuMOvwO5YnEGmsP9+zOAL7xOkDhjEa683wvmurcDgE/EncABy33Kfk0FghuW8fOqXLDc4+iEbDjGAfE+PmKx4VECRwnueE4dIxOzYfPVfk5YxM/n4xh54nJFvi+9PP6G9+cmVw4APlA3QRiVM/ImKif8juEzTnJy/XzvNnqFGBHjhKUdppiw3OEPYgcAPwD+KQ/fOa5YOnpmLgC+sThKGgbv6+fx76NzwHK9McO8Ybmfd6z3+fb4LGWs98ffvRpnLGP7Z+sLSRCy4RSfWO4vtrpywjL+p3BmHniPmuNzo2vUELNh89V/od/+h0XMji42rsgbf/TX7M+YfXLszBCrNJyh813kiLAfmEmI5vj5Z7bR7DEihPSrw4+RN4Qd+BXLTVDIhg52eKgjZxZ6bANftoQO8h/Gn+F/IN4Hdyz34Zvxn7AM6JSxfuN1ggWw9C/FwMhiPmbDqb+j7aac/axC74i1/2I+jA575OxlzIbNV69wGxLvdQafRB+rEUPMhHJcawXUMfP7I+ITnCnfJQXZ1+Pv6dd5fGN8sab181u10awxIsYH1vsfenww0IWE6DfSjp3niHUiB8rogtcHB3jo2hlER87oXbHcR2hp+YQ1MIS+w+AYywqcE+eYDWnIo4r5lA2HYJ9rnT2wjPHRt6G4cPtBakwycI06dlM2bL56QSZYYpOW0fiGzrYo+tmPueP1hrFjVYwjFj/F/2p91weWNnPFGLeW8TwfLS+2MVo/v3UbzRgjQsj2G17MM9D5OEf+TcIbTnXgOxaHNRPcYhHryBm2YaQCxQ3pe+B3Usb8hdfYP8/tRe8Yu39jNhyCKw7sU62Yv2AugcT7TAkeTmxGFjkxGzZfvSIn4DNk5IH0RI24K4UldnjA+CtQGmRsjvkuiuBQ/Oa/p5JZW1Pi57doo9liRAjGBBkjhxXz7KCYw9N0CGdtqRvV/N5oaIyBg3fU5SWuHMQMUDNgKWpT2T/N740Os1988HVUMV9qUzJbnSPmAV0GeCQ0Poxje9SqTBqb2ruvBn5n0kbOsrpobUpOVGrG6h3jxistWqF6Q7ptR99KUWqPW7XRbDHChat7fHZw5LEBYA1gNXCQabM4X5jLyQLrkmZovxi3n4wqBLj/LYbcchHaksCMUGpAUyCO2h4aaMjA2GK+xIYv+J3VyxUH7xizLWrg2B41m6Sx4RSv7qvdB4CHDbwOTAaltk9IUaFZbdL85ixt5EMrVK9I2/Xo2ddSrbZVG80eI1j5ipX9njo2jliziKHDvRA+KFUDMyHaWViLCcSzkYP4G7+dLjNmH+iXiT4+fsc1RtnfIUNkANfsYZMPzHw659Tsl5e0EB9b8Y7fe1GfJeZ9JUHdw83M5NowH4CTS+y5Yl4rPmaBE9memcoDVnuVfoIlz1KlQXNsL8Sr+2r6YtmX0p6eUUa4xIbpW1PIB3pZWlPGgBx7ZMJl5r3NWqGqobePr7F/oFyrbdVGM8cId5vQ08S8LD2UOuQgahUgciu50OGWdHJurejQUZJ9k46U7cbZZ6+9h0esD6XIfmIJKE4s2AehUnPa9nYfHOO98an2nIGsDVA+cmplx44SaLhyjPR29Ef8HV+h4+L8Xa4Nf+Lv8wy5zr7GdxzRpm9bBgmK1h7C5oDVT9yx+mDWiqeNsf99AbvGZ0pe3Ve7yQi3Eof03a0ptWFAL9TkORgX3HPnlEmuSea16NvauNlSqHILSeutIS3sv8bfbtVGs8YIru7JsfkUMe86j+/HhfhKsLkBnBdY08G5y7a1v0tRVXuUdojb3j3ry3PPtltV4iA+p2PgTNJ3b7fA5yFcQc/+zZ0A5f6uJKcmeOgoEd5sW3ds9hTzMpvI36AI8ZUVlBPyXFu6wb88X+LsKShykY6x5mi5havXFhtpS+71coyfxXdD91ZjS2QPvtr1WdJvuhU5Woq2GhvmdadsybUbt2Y+H6Dlv2viEsVkCS36tnYLV0uhyrZr6QNa2X+NHW7ZRjPGiE/PNXcX827JPBqvtuPpbGuyPbnLtsDvJ4RzaZXtqVlqdYVua2EB/C495p777vlcDja3Xek0cuBWDDc45kxcODZKxleLzHxJn3zA73x6iXm37BX7jpOKlCPMsWHanc8ZlYxjJgxyaZV1aeVUe22xkYHc7cdr4HP2g9uuJTbs8uq+Wlbb+Bf4O1kJplVgrrVh+u5UG/u2ELnI5540e+qpIUpo0be1ibBWQpVtWzLOQ7S0/xqttmUbzRYj3uC3m65insvCIaP1CT6XFmK+9GVJLWblz0Yui13wV9S3yvTIgOM6bems5exYDjb3OnKFALfwfOJvtkeb8ZHXNMtDsCxD6QuAPcS8XE3x1fQOOXzfdaXsT5ah9FHi7GuyeiPRa4sNhZKvfGDo5VQhsdZCzL+6r5Y+MCYgpPiu9dktbFgr5uVKR+y7OWU5W6z4bEkLoUrfqHkPRQ4t7X9rMV/aRjPFCGorXxt3E/PyxKEXe0gnExoAtWK+ZNmWpBzuaMhZtm8vdcsMvTxn6EEpt+1kgKrJzHNFQD78ecDvAKEd0NogNQK875SttHJMbtUN37iRy66hNtTa8B1xeysV87UCcwR6bLGJTa4p8NyJo1tNSlLb1nvw1bLNY+JZtnPNakwrGy4R8zE7zZmstEjobUkLoXpHfWUgl9b2v7WYL22jWWIE9V3KRpuLeQq6mLHGBhOpNeSSZVsyS4AgbHPfvcrlz9oXTsgA4QZe+W/uoJNi2+3PHIOisPFldOS40/TdTGL+C6sx+w624Y/zeSmaJXMovqOxYU7s3xG+PymC+FnKL8ziqGP02mIjn/lwJwn8t14Tch978NVym01qJUH661Ja2XDJNpuYKNNOagAT81x5bv2sTGv731LM17TRLDGCDyiH4qOrufl59biRGdLQyWRmvjarF4LCL/fva5/MbrGXKqcTZHYllbkNCWEt0mG77RN7yJUi3xectAbF88cmJFLUptqwphLGs6vZ/Cs4aoSAdPahig5y3IXElMaG3dUj7ZFy/KWVMEaqZkMn3XLCKUWl2z6xh9xiL2SpDYp78dUacS2/N4INa8W8JkHnfi8l5jn+S2jRt1tWs+kl5HvY/1ZivraNZokRpQU3qrL0saUYiZxJhH6Q4q1EfPI6SpZtazK2pcKkphM0SyzSgGvfyheaFPDf3AAkJwA+J889wakVA7ZtLMDJ30q1YY0DenY1m5RTkNfDz2r6WU6+Qv2iycRpbFgjqqRP4Wep/k35oNj1tLDhFlvaelSwiNljyH6k//AJUa0N+9iTr5b7lGPw3DXPArSyYQo8zaoJf1OzB19jIzWTxBZ9u1U1m15Cnudubf81Wm3LNpolRqSSh1KX3cXnVVuzNLNzuR3jHjlXTaUCBpeSZdualwlsUc1GTqBigYXfqck2SNHoXmPIkTPrFgpg2qoy/F4sEMrxlzLymhfObFXNJoRmopNDzHETjoWYCKuxYd/1aNus5oUzo1SzoR9qvcVGilg3+IbEZir5UlMZak++Woql0DVLf17jI1rZMKCv+iETOqF2kds1UjZS82K/Fn27RTWbM9IilS9pLKGH/ddWldqijV4hRpAue+blDC7knDkwNA8saEpn+YjtrU7Bwd77TXwt4f2GBqbsF/e+bljr/KcGglxqCz0cIw1IOu5U8EoZsryH0LkYLDUC6FP5vRlIifkD1odM+WbGGKFqRUQ+JJUSUqU2LClx9qX2Pwr0kxqRe8Fafzh1z9Im3e/6kizStkP+RWvDPvbmq1PZ69ADiFvaMF8AlUIzEeG41oj00kneKOQKVRY6iNkCH4aU39na/vm3tTXbe7ZR6HdnjhFEI+ZzdN7/cLbvcyJv4t80DviO/OVXDr7Shz35sOFMMEP1DX9HhcrbSUPViOBQVkXO9jlbli+VSmURvhFfpSG8j1AJKm15u5yl4xlIiXm31rS2ioQv+8GsyDd0GdESG3bJFfM12z5GgYFV08ayakkqYxUSXDIg0F5llazUmNHasO9a9uirfUJClv11/21LG6bw12xpoO35YpHs79S5arZujILs65TvkvHyM3Kw3yQj2H+pn39WG7m8QowgKTGfq/P+xx0AZ6wzx2/kZW/Y0TlLB+yk3MACrCKv9sGXLbhgfcPuG5Z2P2PNqvja3X2RSaqtZd9KQc3PaEAMFl+J8xE+ra3hJs59xTq+vqAPThxXPfYkbkFKzMvl/dj3JMygMdtzfnz2A11mkJTYsEuumP9Cmf2Pgpyca6AA1LaTXB1lv8h3gxwfB21K4w9zbFj+zZ59NRMsGh+2pQ2zrXPe5ExBxVjEPc6f0E8KapMAW+K+6+UL8dVp930pscNNyo1g/yV+/plt5DJ7jJCkxHyuzvvDBW32B+VmfLifueT3Yi/mmQV3L+gZ8fvh92ngqXs/YGmnD6xLexwg/P935LU/B5s2MB/we3wxYGjhtb8K7MOYE6dd5GwvOqHNHtKSrK0kx4fEXq4xC+xPbXvLPZzabOYV67YNTvgZ4O+PI6e/c22Y12C+Wu/DtrTh3Myr66OvyOtnluKbkdznLHKfwXL/fgT7B/L8/LPbSPIKMUIi+z9037k6rwsXPM9p564cvBKcJeeieahLwzueI7ApOmZ9q2At73j+9qJn2vAdc7wRtAfMoOa2s8zc1AioZ9kwsG9fvYUNH6HbX9+CK9q/KGkPbG3/z/TzNew5RpTqvGZ8oP+SyA3z7b9sBbfPlDhqzvRavJBKu6RXAx8g2yMnbBckn2HDmioHr8wdZcG41YT8WTa8Z1+9pQ2/of/WF1a7m3mv/FZsbf/Ac/x8DXuOETU6r/lF9DJw7sfacweXtq2mtrAWBqpe/fAGEwFbjfHeNkwhuVcRcEf5JFVb/1xD73G2Z1+9tQ0DdeNMA7dpGnmMYv+9/XwNe44RQ/ULH8hovT2CN7nXbRcXlAeHVjWR3esJVaup4QR9FaVX5Irt772XDQNLQJrxYcgWnFCXbWk5IQf62fDeffUINgwsfdvD1t4wdlZ3VEaz/55+voY9x4gandeFAxZH0mqQHLB08GiDbhZkScqWbXhCWzFwwtLPIwTCvdPahoElI7VXJ11L6RsYU7S2YfPVY3FHW5u7wjLyW9DL/nv4+RosRgxKywBhAq8cPmHe40G0lv0y1IzUAGD9OwqyokqPTHrLc5mvHguz4fnpaf/odM4SbHwZhmEYhmEYhmEYhmEYhmEYhmEYhmEYhmEYhmEYhmEYhmEYhmEYhmEYhmEYxkz8B+9eXgSCQ2hrAAAAAElFTkSuQmCC\" alt=\"sigma(7) = 8, phi(8) = 4, sigma(4) = 7, phi(7) = 6, sigma(6) = 12, phi(12) = 4, \" style=\"width: 377.5px; height: 19px;\" width=\"377.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.2167px 7.91667px; transform-origin: 13.2167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e etc.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 117.458px 7.91667px; transform-origin: 117.458px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand the pattern 7, 6, 12, 4 will repeat. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 258.3px 7.91667px; transform-origin: 258.3px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eOscillating behavior is plausible because the sum of divisors is always greater than \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 120.45px 7.91667px; transform-origin: 120.45px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and the totient is always smaller than \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 360.958px 7.91667px; transform-origin: 360.958px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Furthermore, because the totient has a minimum value and the sum of divisors has a maximum value, with enough iterations the sequence would have to hit a repeating pattern.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 383.4px 7.91667px; transform-origin: 383.4px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes an initial seed and returns the repeating pattern and the index of the sequence where the pattern begins. With an initial seed of 7, the sequence would be 7, 8, 4, 7, 6, 12, 4, 7, 6, 12,… Therefore, the repeating pattern is [7 6 12 4] and the start index is 3.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [q,n0] = sigPhi(n)\r\n% n = initial seed\r\n% q = vector of repeating pattern\r\n% n0 = index where the repeating pattern starts (counting the initial seed as index 1)\r\nend","test_suite":"%%\r\nn = 2;\r\nq_correct = [2 3];\r\nn0_correct = 1;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 3;\r\nq_correct = [2 3];\r\nn0_correct = 3;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 7;\r\nq_correct = [4 7 6 12];\r\nn0_correct = 3;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 12;\r\nq_correct = [12 28];\r\nn0_correct = 1;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 28;\r\nq_correct = [24 60 16 31 30 72];\r\nn0_correct = 3;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 101;\r\nq_correct = [72 195 96 252];\r\nn0_correct = 7;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 127;\r\nq_correct = [96 252 72 195];\r\nn0_correct = 7;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 256;\r\nq_correct = [432 1240 480 1512];\r\nn0_correct = 3;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 777;\r\nq_correct = [576 1651 1512 4800 1280 3066 864 2520];\r\nn0_correct = 3;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 1111;\r\nq_correct = [432 1240 480 1512];\r\nn0_correct = 7;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 5555;\r\nq_correct = [10368 30855 14080 36792];\r\nn0_correct = 23;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 7777;\r\nq_correct = [3024 9920 3840 12264 3456 10200 2560 6138 1800 6045 2880 9906];\r\nn0_correct = 11;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 11111;\r\nq_correct = [3024 9920 3840 12264 3456 10200 2560 6138 1800 6045 2880 9906];\r\nn0_correct = 11;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 77777;\r\nq_correct = [10368 30855 14080 36792];\r\nn0_correct = 27;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 123456;\r\nq_correct = [184320 638898 196560 833280];\r\nn0_correct = 21;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 666666;\r\nq_correct = [1658880 5946666 1801800 8124480];\r\nn0_correct = 39;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 7777777;\r\nq_correct = [191102976000 715162215924 207622711296 859454668800 178362777600 757256331104 283740364800 1100946774480 233003796480 1053092362140 221908377600 1035248323200 204838502400 888208962000 214695936000 952677206208 237283098624 859638312960 185794560000 792731088600 178886400000 749337039360 150493593600 639777817224 152374763520 626874655824 202491394560 925865740800 167215104000 715161022368 219847799808 880002352320 161864220672 609720615224 247328774784 987821856000];\r\nn0_correct = 161;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))","published":true,"deleted":false,"likes_count":3,"comments_count":4,"created_by":46909,"edited_by":46909,"edited_at":"2022-11-28T04:11:02.000Z","deleted_by":null,"deleted_at":null,"solvers_count":17,"test_suite_updated_at":"2021-05-10T14:27:43.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-09T19:28:03.000Z","updated_at":"2026-01-14T13:15:59.000Z","published_at":"2021-05-09T19:36:55.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46898\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 46898\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e deals with the sum of divisors function, denoted by \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"sigma(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, while \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/656\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 656\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e deals with the totient function, denoted by \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"phi(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\varphi(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The sum of divisors is straightforward: for example, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"sigma(12) = 1+2+3+4+6+12 = 28\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma(12) = 1+2+3+4+6+12 = 28\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The totient of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e counts the numbers less than \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e that are relatively prime to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. For example \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"phi(12) = 4\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\varphi(12) = 4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e because the greatest common divisor of 12 and four numbers (1, 5, 7, 11) is 1. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhat happens if you repeatedly apply the two functions, starting with the sum of divisors and alternating? For example, start with 7. Then \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"sigma(7) = 8, phi(8) = 4, sigma(4) = 7, phi(7) = 6, sigma(6) = 12, phi(12) = 4, \\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma(7) = 8, \\\\varphi(8) = 4, \\\\sigma(4) = 7, \\\\varphi(7) = 6, \\\\sigma(6) = 12, \\\\varphi(12) = 4,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e etc.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eand the pattern 7, 6, 12, 4 will repeat. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOscillating behavior is plausible because the sum of divisors is always greater than \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and the totient is always smaller than \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Furthermore, because the totient has a minimum value and the sum of divisors has a maximum value, with enough iterations the sequence would have to hit a repeating pattern.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes an initial seed and returns the repeating pattern and the index of the sequence where the pattern begins. With an initial seed of 7, the sequence would be 7, 8, 4, 7, 6, 12, 4, 7, 6, 12,… Therefore, the repeating pattern is [7 6 12 4] and the start index is 3.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":43755,"title":"Divide the Least Common Multiple by the Greatest Common Divisor of two numbers","description":"Divide the Least Common Multiple by the Greatest Common Divisor of two numbers. For example, for x=12345 and y=54321, the answer would be\r\n\r\n 223530915/3 =\r\n 74510305 (ANSWER)","description_html":"\u003cp\u003eDivide the Least Common Multiple by the Greatest Common Divisor of two numbers. For example, for x=12345 and y=54321, the answer would be\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003e223530915/3 =\r\n74510305 (ANSWER)\r\n\u003c/pre\u003e","function_template":"function z = your_fcn_name(x,y)\r\n y = x;\r\nend","test_suite":"%%\r\nx = 12345;\r\ny = 54321;\r\ny_correct = 74510305;\r\nassert(isequal(your_fcn_name(x,y),y_correct))\r\n%%\r\nx = 12;\r\ny = 54;\r\ny_correct = 18;\r\nassert(isequal(your_fcn_name(x,y),y_correct))\r\n%%\r\nx = 1;\r\ny = 1;\r\ny_correct = 1;\r\nassert(isequal(your_fcn_name(x,y),y_correct))\r\n%%\r\nx = 987654321;\r\ny = x;\r\ny_correct = 1;\r\nassert(isequal(your_fcn_name(x,y),y_correct))\r\n%%\r\nx = 800;\r\ny = 26000;\r\ny_correct = 130;\r\nassert(isequal(your_fcn_name(x,y),y_correct))\r\n%%\r\nx = 65536;\r\ny = 32768;\r\ny_correct = 2;\r\nassert(isequal(your_fcn_name(x,y),y_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":93456,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":53,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2016-12-07T23:00:01.000Z","updated_at":"2026-03-19T18:26:49.000Z","published_at":"2016-12-07T23:00:01.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDivide the Least Common Multiple by the Greatest Common Divisor of two numbers. For example, for x=12345 and y=54321, the answer would be\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[223530915/3 =\\n74510305 (ANSWER)]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":49327,"title":"Divisors","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 21px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 343px 10.5px; transform-origin: 343px 10.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 320px 10.5px; text-align: left; transform-origin: 320px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eFind the divisors of a given number. But don't return the number itself, neither 1.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = divisores(n)\r\n y = n;\r\nend","test_suite":"%%\r\nx = 20;\r\ny_correct = [2,4,5,10];\r\nassert(isequal(divisores(x),y_correct))\r\n%%\r\nx = 30;\r\ny_correct = [2,3,5,6,10,15];\r\nassert(isequal(divisores(x),y_correct))\r\n%%\r\nx = 77;\r\ny_correct = [7,11];\r\nassert(isequal(divisores(x),y_correct))\r\n%%\r\nx = 101; %dalmatians\r\ny_correct = [];\r\nassert(isequal(divisores(x),y_correct))\r\n%%\r\nx = 777;\r\ny_correct = [3,7,21,37,111,259];\r\nassert(isequal(divisores(x),y_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":698530,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":40,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-12-23T11:05:39.000Z","updated_at":"2026-02-28T08:27:11.000Z","published_at":"2020-12-31T01:21:53.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFind the divisors of a given number. But don't return the number itself, neither 1.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":44748,"title":"Amicable numbers","description":"Test whether two numbers are \u003chttps://en.wikipedia.org/wiki/Amicable_numbers amicable\u003e, meaning that the sum of the proper divisors of each number is equal to the other number.\r\n\r\n\r\nExample: 220 and 284 are amicable numbers because the proper divisors of 220 are 1,2,4,5,10,11,20,22,44,55,110 and their sum is 284, while the proper divisors of 284 are 1,2,4,71,142 and their sum is 220.","description_html":"\u003cp\u003eTest whether two numbers are \u003ca href = \"https://en.wikipedia.org/wiki/Amicable_numbers\"\u003eamicable\u003c/a\u003e, meaning that the sum of the proper divisors of each number is equal to the other number.\u003c/p\u003e\u003cp\u003eExample: 220 and 284 are amicable numbers because the proper divisors of 220 are 1,2,4,5,10,11,20,22,44,55,110 and their sum is 284, while the proper divisors of 284 are 1,2,4,71,142 and their sum is 220.\u003c/p\u003e","function_template":"function y = amicable(m,n)\r\n y = false;\r\nend","test_suite":"%%\r\nm = 220; n = 284;\r\ny_correct = true;\r\nassert(isequal(amicable(m,n),y_correct))\r\n\r\n%%\r\nm = 220; n = 504;\r\ny_correct = false;\r\nassert(isequal(amicable(m,n),y_correct))\r\n\r\n%%\r\nm = 2620; n = 2924;\r\ny_correct = true;\r\nassert(isequal(amicable(m,n),y_correct))\r\n\r\n%%\r\nm = 5020; n = 5564;\r\ny_correct = true;\r\nassert(isequal(amicable(m,n),y_correct))\r\n\r\n%%\r\nm = 2924; n = 5020;\r\ny_correct = false;\r\nassert(isequal(amicable(m,n),y_correct))","published":true,"deleted":false,"likes_count":3,"comments_count":4,"created_by":254267,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":78,"test_suite_updated_at":"2018-10-22T18:10:27.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2018-10-22T17:57:52.000Z","updated_at":"2026-03-16T15:34:17.000Z","published_at":"2018-10-22T18:02:09.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTest whether two numbers are\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Amicable_numbers\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eamicable\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, meaning that the sum of the proper divisors of each number is equal to the other number.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample: 220 and 284 are amicable numbers because the proper divisors of 220 are 1,2,4,5,10,11,20,22,44,55,110 and their sum is 284, while the proper divisors of 284 are 1,2,4,71,142 and their sum is 220.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":45794,"title":"*Prime number check 2 (in construction)","description":"Another way to see if a number is prime is to count the number of factors. For example,\r\n\r\n the number 4 has 2 factors, [ 2 4 ]\r\n the number 16 has 4 factors, [ 2 4 8 16 ]\r\n the number 7 has 1 factor, [ 7 ]\r\n\r\nThus, if a number has only one factor, it is prime. We ignore the factor 1 as it is trivial. \r\n\r\nYour function should create a vector containing all the factors of any number x.","description_html":"\u003cp\u003eAnother way to see if a number is prime is to count the number of factors. For example,\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003ethe number 4 has 2 factors, [ 2 4 ]\r\nthe number 16 has 4 factors, [ 2 4 8 16 ]\r\nthe number 7 has 1 factor, [ 7 ]\r\n\u003c/pre\u003e\u003cp\u003eThus, if a number has only one factor, it is prime. We ignore the factor 1 as it is trivial.\u003c/p\u003e\u003cp\u003eYour function should create a vector containing all the factors of any number x.\u003c/p\u003e","function_template":"function y = your_fcn_name(x)\r\n y = x;\r\nend","test_suite":"%%\r\nx = 16;\r\ny_correct = [ 2 4 8 16 ];\r\nassert(isequal(your_fcn_name(x),y_correct))\r\n%%\r\nx = 23;\r\ny_correct = [ 23 ];\r\nassert(isequal(your_fcn_name(x),y_correct))\r\n%%\r\nx = 100;\r\ny_correct = [ 2 4 5 10 20 25 50 100 ];\r\nassert(isequal(your_fcn_name(x),y_correct))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":428668,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":29,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-06-07T20:00:06.000Z","updated_at":"2025-08-03T17:44:27.000Z","published_at":"2020-06-07T20:00:06.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAnother way to see if a number is prime is to count the number of factors. For example,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[the number 4 has 2 factors, [ 2 4 ]\\nthe number 16 has 4 factors, [ 2 4 8 16 ]\\nthe number 7 has 1 factor, [ 7 ]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThus, if a number has only one factor, it is prime. We ignore the factor 1 as it is trivial.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYour function should create a vector containing all the factors of any number x.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":1933,"title":"That's some divisor you've got there...","description":"Given a positive integer x, calculate the sum of all of the divisors of the number. Please include the number itself in your final answer. For example, 1, 3, 5 and 15 are all the proper divisors of 15, so sum_divisors(15)=1+3+5+15, which is 24.\r\n\r\nYour score will be based on how quickly your script solves the problems in the test suite, rather than the typical Cody size. Good luck!","description_html":"\u003cp\u003eGiven a positive integer x, calculate the sum of all of the divisors of the number. Please include the number itself in your final answer. For example, 1, 3, 5 and 15 are all the proper divisors of 15, so sum_divisors(15)=1+3+5+15, which is 24.\u003c/p\u003e\u003cp\u003eYour score will be based on how quickly your script solves the problems in the test suite, rather than the typical Cody size. Good luck!\u003c/p\u003e","function_template":"function y = sum_divisors(x)\r\n y = x;\r\nend","test_suite":"%%\r\nt_in=clock;\r\n\r\nx = 1; y_correct = 1; yours=sum_divisors(x);\r\nassert(isequal(yours,y_correct))\r\nt_out=etime(clock,t_in)*1000;\r\nfprintf('Actual Time = %.0f msec\\n',t_out)\r\n\r\nx = 2; y_correct = 3; yours=sum_divisors(x);\r\nassert(isequal(yours,y_correct))\r\nt_out=etime(clock,t_in)*1000;\r\nfprintf('Actual Time = %.0f msec\\n',t_out)\r\n\r\nx = 120; y_correct = 360; yours=sum_divisors(x);\r\nassert(isequal(yours,y_correct))\r\nt_out=etime(clock,t_in)*1000;\r\nfprintf('Actual Time = %.0f msec\\n',t_out)\r\n\r\n% Perfect Number!\r\nx = 33550336; y_correct = 67100672; yours=sum_divisors(x);\r\nassert(isequal(yours,y_correct))\r\nt_out=etime(clock,t_in)*1000;\r\nfprintf('Actual Time = %.0f msec\\n',t_out)\r\n\r\nx = 223092870; y_correct = 836075520; yours=sum_divisors(x);\r\nassert(isequal(yours,y_correct))\r\nt_out=etime(clock,t_in)*1000;\r\nfprintf('Actual Time = %.0f msec\\n',t_out)\r\n\r\nx = 4294967295; y_correct = 7304603328; yours=sum_divisors(x);\r\nassert(isequal(yours,y_correct))\r\nt_out=etime(clock,t_in)*1000;\r\nfprintf('Actual Time = %.0f msec\\n',t_out)\r\n\r\nx=arrayfun(@(y) sum_divisors(y),1:20000);assert(isequal(sum(x),329004151));\r\nt_out=etime(clock,t_in)*1000;\r\nassert(all(x\u003e0));\r\nfprintf('Actual Time = %.0f msec\\n',t_out)\r\n\r\nx=arrayfun(@(p) sum_divisors(p),primes(200000));assert(isequal(sum(x),1709618797));\r\nt_out=etime(clock,t_in)*1000;\r\nassert(all(x\u003e0));\r\n\r\nudx=unique(diff(x));\r\nassert(isequal(numel(udx),39))\r\nassert(isequal(sum_divisors(max(udx)),132))\r\n\r\nt2=min(100000,t_out);\r\nfprintf('Actual Time = %.0f msec\\n',t_out)\r\nfeval(@assignin,'caller','score',floor(t2));","published":true,"deleted":false,"likes_count":4,"comments_count":0,"created_by":1615,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":36,"test_suite_updated_at":"2018-08-22T14:05:40.000Z","rescore_all_solutions":false,"group_id":38,"created_at":"2013-10-11T15:21:25.000Z","updated_at":"2025-03-08T05:05:07.000Z","published_at":"2013-10-11T15:21:25.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a positive integer x, calculate the sum of all of the divisors of the number. Please include the number itself in your final answer. For example, 1, 3, 5 and 15 are all the proper divisors of 15, so sum_divisors(15)=1+3+5+15, which is 24.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYour score will be based on how quickly your script solves the problems in the test suite, rather than the typical Cody size. Good luck!\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":60935,"title":"Compute the antisigma function","description":"The sum of divisors function, or , is the sum of the numbers that divide . The antisigma function is the sum of numbers less than that do not divide . For example, antisigma(15) = 2+4+6+7+8+9+10+11+12+13+14 = 96. \r\nWrite a function that computes the antisigma function. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 72px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 36px; transform-origin: 408px 36px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 99.175px 8px; transform-origin: 99.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe sum of divisors function, or \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"30.5\" height=\"18.5\" alt=\"sigma(n)\" style=\"width: 30.5px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 120.175px 8px; transform-origin: 120.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, is the sum of the numbers that divide \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 145.208px 8px; transform-origin: 145.208px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The antisigma function is the sum of numbers less than \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 57.175px 8px; transform-origin: 57.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that do not divide \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 219.358px 8px; transform-origin: 219.358px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. For example, antisigma(15) = 2+4+6+7+8+9+10+11+12+13+14 = 96. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 168.283px 8px; transform-origin: 168.283px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that computes the antisigma function. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = antisigma(n)\r\n y = n-sum(n);\r\nend","test_suite":"%%\r\nassert(isequal(antisigma(0),0))\r\n\r\n%%\r\nassert(isequal(antisigma(1),0))\r\n\r\n%%\r\nassert(isequal(antisigma(15),96))\r\n\r\n%%\r\nassert(isequal(antisigma(66),2067))\r\n\r\n%%\r\nassert(isequal(antisigma(214),22681))\r\n\r\n%%\r\nassert(isequal(antisigma(651),211202))\r\n\r\n%%\r\nassert(isequal(antisigma(2030),2057145))\r\n\r\n%%\r\nassert(isequal(antisigma(4306),9266509))\r\n\r\n%%\r\nassert(isequal(antisigma(6771),22917182))\r\n\r\n%%\r\nassert(isequal(antisigma(8888),39484356))\r\n\r\n%%\r\nassert(isequal(antisigma(9999),49979088))\r\n\r\n%%\r\nassert(isequal(antisigma(45364),1028882242))\r\n\r\n%%\r\nassert(isequal(antisigma(351433),61752399801))\r\n\r\n%%\r\nassert(isequal(antisigma(7665534),29380192820025))\r\n\r\n%%\r\nassert(isequal(antisigma(88432534),3910156446398041))\r\n\r\n%%\r\np = primes(1e5);\r\nq = p(randi(length(p)));\r\nassert(isequal(antisigma(q),polyval([1/2 -1/2 -1],q)))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2025-06-12T01:47:32.000Z","deleted_by":null,"deleted_at":null,"solvers_count":8,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-06-12T01:47:03.000Z","updated_at":"2025-10-14T02:58:47.000Z","published_at":"2025-06-12T01:47:32.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe sum of divisors function, or \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"sigma(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, is the sum of the numbers that divide \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The antisigma function is the sum of numbers less than \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e that do not divide \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. For example, antisigma(15) = 2+4+6+7+8+9+10+11+12+13+14 = 96. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that computes the antisigma function. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":42791,"title":"Number of divisors of a given number","description":"Given a Number n, return the number of his divisors without listing them\r\n\r\nexample: \r\n\r\nn=14\r\n; Divisors={1,7,2,14} ; y=4\r\n\r\nn=68\r\n; Divisors={1,2,34,17,4,68} ; y=6","description_html":"\u003cp\u003eGiven a Number n, return the number of his divisors without listing them\u003c/p\u003e\u003cp\u003eexample:\u003c/p\u003e\u003cp\u003en=14\r\n; Divisors={1,7,2,14} ; y=4\u003c/p\u003e\u003cp\u003en=68\r\n; Divisors={1,2,34,17,4,68} ; y=6\u003c/p\u003e","function_template":"function y = your_fcn_name(n)\r\n\r\nend","test_suite":"%%\r\nfiletext = fileread('your_fcn_name.m');\r\nassert(isempty(strfind(filetext, 'sqrt')))\r\nassert(isempty(strfind(filetext, 'for')))\r\n\r\n%%\r\nn= 6880;\r\ny_correct = 24;\r\nassert(isequal(your_fcn_name(n),y_correct))\r\n\r\n%%\r\nn= 5050;\r\ny_correct = 12;\r\nassert(isequal(your_fcn_name(n),y_correct))\r\n\r\n%%\r\nn= 188325;\r\ny_correct = 36;\r\nassert(isequal(your_fcn_name(n),y_correct))\r\n\r\n\r\n%%\r\nn= 76576500;\r\ny_correct = 576;\r\nassert(isequal(your_fcn_name(n),y_correct))\r\n\r\n%%\r\nn= 74;\r\ny_correct = 4;\r\nassert(isequal(your_fcn_name(n),y_correct))\r\n\r\n\r\n%%\r\nn=14^8 ;\r\ny_correct = 81;\r\nassert(isequal(your_fcn_name(n),y_correct))\r\n\r\n\r\n%%\r\nn=314721 ;\r\ny_correct = 27;\r\nassert(isequal(your_fcn_name(n),y_correct))","published":true,"deleted":false,"likes_count":2,"comments_count":6,"created_by":17228,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":66,"test_suite_updated_at":"2016-04-01T06:25:39.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2016-03-31T12:09:31.000Z","updated_at":"2025-12-03T20:02:50.000Z","published_at":"2016-03-31T12:09:31.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a Number n, return the number of his divisors without listing them\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eexample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003en=14 ; Divisors={1,7,2,14} ; y=4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003en=68 ; Divisors={1,2,34,17,4,68} ; y=6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":59841,"title":"Count the ways to write 1/n as the sum of two unit fractions","description":"The number 1/2 can be written as 1/3+1/6 and 1/4+1/4, and the number 1/9 can be written as 1/18+1/18, 1/12+1/36, and 1/10+1/90. That is, 1/2 can be written as the sum of two unit fractions (those with a numerator of 1) in two ways, and 1/9 can be written as the sum of two unit fractions in three ways. \r\nWrite a function to count the ways can be written as the sum of two unit fractions.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 93px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 46.5px; transform-origin: 407px 46.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 374.208px 8px; transform-origin: 374.208px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe number 1/2 can be written as 1/3+1/6 and 1/4+1/4, and the number 1/9 can be written as 1/18+1/18, 1/12+1/36, and 1/10+1/90. That is, 1/2 can be written as the sum of two unit fractions (those with a numerator of 1) in two ways, and 1/9 can be written as the sum of two unit fractions in three ways. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 106.825px 8px; transform-origin: 106.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to count the ways \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"25.5\" height=\"18.5\" alt=\"1/n\" style=\"width: 25.5px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 145.842px 8px; transform-origin: 145.842px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e can be written as the sum of two unit fractions.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = twoUnitFractions(n)\r\n y = 1/n+1/n;\r\nend","test_suite":"%%\r\nn = 2;\r\ny = twoUnitFractions(n);\r\ny_correct = 2;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 9;\r\ny = twoUnitFractions(n);\r\ny_correct = 3;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 85;\r\ny = twoUnitFractions(n);\r\ny_correct = 5;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 327;\r\ny = twoUnitFractions(n);\r\ny_correct = 5;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 461;\r\ny = twoUnitFractions(n);\r\ny_correct = 2;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 824;\r\ny = twoUnitFractions(n);\r\ny_correct = 11;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 1989;\r\ny = twoUnitFractions(n);\r\ny_correct = 23;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 5460;\r\ny = twoUnitFractions(n);\r\ny_correct = 203;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 7011;\r\ny = twoUnitFractions(n);\r\ny_correct = 23;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 8610;\r\ny = twoUnitFractions(n);\r\ny_correct = 122;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 9438;\r\ny = twoUnitFractions(n);\r\ny_correct = 68;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 14332;\r\ny = twoUnitFractions(n);\r\ny_correct = 8;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 58110;\r\ny = twoUnitFractions(n);\r\ny_correct = 122;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 254000;\r\ny = twoUnitFractions(n);\r\ny_correct = 95;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 5563449;\r\ny = twoUnitFractions(n);\r\ny_correct = 8;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 77221133;\r\ny = twoUnitFractions(n);\r\ny_correct = 5;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\np = primes(1000);\r\ny = arrayfun(@twoUnitFractions,p);\r\nassert(all(y==2))\r\n\r\n%%\r\np = primes(20);\r\nb = p(randi(length(p)));\r\ne = randi(5);\r\ny = twoUnitFractions(b^e);\r\nassert(isequal(y,e+1))\r\n\r\n%%\r\nfiletext = fileread('twoUnitFractions.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2024-04-05T01:09:45.000Z","deleted_by":null,"deleted_at":null,"solvers_count":7,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2024-04-05T01:09:23.000Z","updated_at":"2025-07-19T12:22:30.000Z","published_at":"2024-04-05T01:09:45.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe number 1/2 can be written as 1/3+1/6 and 1/4+1/4, and the number 1/9 can be written as 1/18+1/18, 1/12+1/36, and 1/10+1/90. That is, 1/2 can be written as the sum of two unit fractions (those with a numerator of 1) in two ways, and 1/9 can be written as the sum of two unit fractions in three ways. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to count the ways \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"1/n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e1/n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e can be written as the sum of two unit fractions.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":2664,"title":"Divisors for big integer","description":"Inspired by Problem 1025 and Project Euler 12.\r\n\r\nGiven n, return the number y of integers that divide N. \r\n\r\nFor example, with n = 10, the divisors are [1 2 5 10], so y = 4.\r\n\r\nIt's easy with normal integer but how to proceed with big number?\r\n\r\n","description_html":"\u003cp\u003eInspired by Problem 1025 and Project Euler 12.\u003c/p\u003e\u003cp\u003eGiven n, return the number y of integers that divide N.\u003c/p\u003e\u003cp\u003eFor example, with n = 10, the divisors are [1 2 5 10], so y = 4.\u003c/p\u003e\u003cp\u003eIt's easy with normal integer but how to proceed with big number?\u003c/p\u003e","function_template":"function y = divisors_Big(x)\r\n y = x;\r\nend","test_suite":"%%\r\nx = 10;\r\ny_correct = 4;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 28;\r\ny_correct = 6;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 28;\r\ny_correct = 6;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 784;\r\ny_correct = 15;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 1452637;\r\ny_correct = 2;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 5452637;\r\ny_correct = 4;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 16452637;\r\ny_correct = 2;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 116452637;\r\ny_correct = 8;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 416452638;\r\ny_correct = 32;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 12250000;\r\ny_correct = 105;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 2031120;\r\ny_correct = 240;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 76576500;\r\ny_correct = 576;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 816452637;\r\ny_correct = 32;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 103672800;\r\ny_correct = 648;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n%%\r\nx = 842161320;\r\ny_correct = 1024;\r\nassert(isequal(divisors_Big(x),y_correct))\r\n\r\n","published":true,"deleted":false,"likes_count":4,"comments_count":0,"created_by":5390,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":240,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2014-11-12T02:00:29.000Z","updated_at":"2026-01-05T00:22:49.000Z","published_at":"2014-11-12T02:28:56.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eInspired by Problem 1025 and Project Euler 12.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven n, return the number y of integers that divide N.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, with n = 10, the divisors are [1 2 5 10], so y = 4.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIt's easy with normal integer but how to proceed with big number?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":46898,"title":"Sum of all the divisors of n","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 21px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 10.5px; transform-origin: 407px 10.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eOutput the sum of all the divisors of a number (n). For example n=10, divisors = 1, 2, 5, 10, output = 18.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = sumDivisors(x)\r\n y = sum(x);\r\nend","test_suite":"%%\r\nx=2.^(10:20);\r\ny=[2047,4095,8191,16383,32767,65535,131071,262143,524287,1048575,2097151];\r\nassert(isequal(sumDivisors(x),y))\r\n%%\r\nx=2^32-1;\r\ny=7304603328;\r\nassert(isequal(sumDivisors(x),y))\r\n%%\r\nx =[ 53394 10330 42975 46272 28754 18090 49236 55098 23047 4972\r\n 59362 63609 2341 2087 25006 44545 16718 16665 54450 3536\r\n 8323 62729 55649 18149 50169 42933 33159 53365 38356 34787\r\n 59860 31810 61211 3026 52115 10657 45815 15960 36027 51064\r\n 41443 52448 44482 6366 12247 7799 58387 60901 60110 61212\r\n 6393 9299 49660 53967 32098 32661 62869 22937 18733 8514\r\n 18252 27641 48702 45537 29202 62898 35863 12885 49624 37279\r\n 35841 60014 25705 20782 42357 22308 9085 16456 49397 30762\r\n 62752 51919 42958 62274 46489 38357 9785 40374 24933 781\r\n 63235 62882 11219 2258 49460 14668 16877 31018 37213 22094];\r\ny=[ 116640 18612 77376 122936 47088 48960 125664 119418 24280 9576\r\n 90576 104832 2342 2088 37512 54720 27048 29376 160797 7812\r\n 10080 63300 60720 18150 76480 62496 50560 69048 68992 35640\r\n 130536 57276 61212 4860 71520 10658 73872 57600 52052 103320\r\n 41444 113400 69696 12744 12616 8520 70400 60902 108216 142856\r\n 8528 9864 112896 71960 52560 49920 62870 22938 22176 20592\r\n 51240 27984 97416 62304 60672 137376 35864 20640 93060 40680\r\n 51520 92568 31752 31176 64576 61488 11520 35910 50496 66690\r\n 129276 59344 65952 127008 46490 42294 12480 87516 33248 864\r\n 75888 98496 12096 3390 103908 27160 19296 50148 43200 33144];\r\nassert(isequal(sumDivisors(x),y))\r\n\r\n","published":true,"deleted":false,"likes_count":4,"comments_count":0,"created_by":145982,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":26,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-10-18T03:07:30.000Z","updated_at":"2025-11-29T15:06:14.000Z","published_at":"2020-10-18T03:27:54.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOutput the sum of all the divisors of a number (n). For example n=10, divisors = 1, 2, 5, 10, output = 18.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51715,"title":"Iterate the sum of divisors and totient","description":"","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 339px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 169.5px; transform-origin: 407px 169.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46898\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003eCody Problem 46898\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 160.25px 7.91667px; transform-origin: 160.25px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e deals with the sum of divisors function, denoted by \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"sigma(n)\" style=\"width: 30.5px; height: 19px;\" width=\"30.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.7833px 7.91667px; transform-origin: 21.7833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, while \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/656\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003eCody Problem 656\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 46.675px 7.91667px; transform-origin: 46.675px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e deals with the totient function, denoted by \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"phi(n)\" style=\"width: 31.5px; height: 19px;\" width=\"31.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 164.125px 7.91667px; transform-origin: 164.125px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The sum of divisors is straightforward: for example, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"sigma(12) = 1+2+3+4+6+12 = 28\" style=\"width: 227.5px; height: 19px;\" width=\"227.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The totient of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 95.2917px 7.91667px; transform-origin: 95.2917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e counts the numbers less than \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 84.4px 7.91667px; transform-origin: 84.4px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that are relatively prime to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 44.725px 7.91667px; transform-origin: 44.725px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. For example \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"phi(12) = 4\" style=\"width: 65px; height: 19px;\" width=\"65\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68.5083px 7.91667px; transform-origin: 68.5083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e because the greatest common divisor of 12 and four numbers (1, 5, 7, 11) is 1. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 384px 7.91667px; transform-origin: 384px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWhat happens if you repeatedly apply the two functions, starting with the sum of divisors and alternating? For example, start with 7. Then \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"sigma(7) = 8, phi(8) = 4, sigma(4) = 7, phi(7) = 6, sigma(6) = 12, phi(12) = 4, \" style=\"width: 377.5px; height: 19px;\" width=\"377.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.2167px 7.91667px; transform-origin: 13.2167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e etc.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 117.458px 7.91667px; transform-origin: 117.458px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand the pattern 7, 6, 12, 4 will repeat. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 258.3px 7.91667px; transform-origin: 258.3px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eOscillating behavior is plausible because the sum of divisors is always greater than \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 120.45px 7.91667px; transform-origin: 120.45px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and the totient is always smaller than \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 360.958px 7.91667px; transform-origin: 360.958px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Furthermore, because the totient has a minimum value and the sum of divisors has a maximum value, with enough iterations the sequence would have to hit a repeating pattern.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 383.4px 7.91667px; transform-origin: 383.4px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes an initial seed and returns the repeating pattern and the index of the sequence where the pattern begins. With an initial seed of 7, the sequence would be 7, 8, 4, 7, 6, 12, 4, 7, 6, 12,… Therefore, the repeating pattern is [7 6 12 4] and the start index is 3.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [q,n0] = sigPhi(n)\r\n% n = initial seed\r\n% q = vector of repeating pattern\r\n% n0 = index where the repeating pattern starts (counting the initial seed as index 1)\r\nend","test_suite":"%%\r\nn = 2;\r\nq_correct = [2 3];\r\nn0_correct = 1;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 3;\r\nq_correct = [2 3];\r\nn0_correct = 3;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 7;\r\nq_correct = [4 7 6 12];\r\nn0_correct = 3;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 12;\r\nq_correct = [12 28];\r\nn0_correct = 1;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 28;\r\nq_correct = [24 60 16 31 30 72];\r\nn0_correct = 3;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 101;\r\nq_correct = [72 195 96 252];\r\nn0_correct = 7;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 127;\r\nq_correct = [96 252 72 195];\r\nn0_correct = 7;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 256;\r\nq_correct = [432 1240 480 1512];\r\nn0_correct = 3;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 777;\r\nq_correct = [576 1651 1512 4800 1280 3066 864 2520];\r\nn0_correct = 3;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 1111;\r\nq_correct = [432 1240 480 1512];\r\nn0_correct = 7;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 5555;\r\nq_correct = [10368 30855 14080 36792];\r\nn0_correct = 23;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 7777;\r\nq_correct = [3024 9920 3840 12264 3456 10200 2560 6138 1800 6045 2880 9906];\r\nn0_correct = 11;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 11111;\r\nq_correct = [3024 9920 3840 12264 3456 10200 2560 6138 1800 6045 2880 9906];\r\nn0_correct = 11;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 77777;\r\nq_correct = [10368 30855 14080 36792];\r\nn0_correct = 27;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 123456;\r\nq_correct = [184320 638898 196560 833280];\r\nn0_correct = 21;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 666666;\r\nq_correct = [1658880 5946666 1801800 8124480];\r\nn0_correct = 39;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 7777777;\r\nq_correct = [191102976000 715162215924 207622711296 859454668800 178362777600 757256331104 283740364800 1100946774480 233003796480 1053092362140 221908377600 1035248323200 204838502400 888208962000 214695936000 952677206208 237283098624 859638312960 185794560000 792731088600 178886400000 749337039360 150493593600 639777817224 152374763520 626874655824 202491394560 925865740800 167215104000 715161022368 219847799808 880002352320 161864220672 609720615224 247328774784 987821856000];\r\nn0_correct = 161;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))","published":true,"deleted":false,"likes_count":3,"comments_count":4,"created_by":46909,"edited_by":46909,"edited_at":"2022-11-28T04:11:02.000Z","deleted_by":null,"deleted_at":null,"solvers_count":17,"test_suite_updated_at":"2021-05-10T14:27:43.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-09T19:28:03.000Z","updated_at":"2026-01-14T13:15:59.000Z","published_at":"2021-05-09T19:36:55.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46898\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 46898\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e deals with the sum of divisors function, denoted by \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"sigma(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, while \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/656\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 656\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e deals with the totient function, denoted by \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"phi(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\varphi(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The sum of divisors is straightforward: for example, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"sigma(12) = 1+2+3+4+6+12 = 28\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma(12) = 1+2+3+4+6+12 = 28\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The totient of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e counts the numbers less than \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e that are relatively prime to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. For example \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"phi(12) = 4\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\varphi(12) = 4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e because the greatest common divisor of 12 and four numbers (1, 5, 7, 11) is 1. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhat happens if you repeatedly apply the two functions, starting with the sum of divisors and alternating? For example, start with 7. Then \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"sigma(7) = 8, phi(8) = 4, sigma(4) = 7, phi(7) = 6, sigma(6) = 12, phi(12) = 4, \\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma(7) = 8, \\\\varphi(8) = 4, \\\\sigma(4) = 7, \\\\varphi(7) = 6, \\\\sigma(6) = 12, \\\\varphi(12) = 4,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e etc.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eand the pattern 7, 6, 12, 4 will repeat. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOscillating behavior is plausible because the sum of divisors is always greater than \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and the totient is always smaller than \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Furthermore, because the totient has a minimum value and the sum of divisors has a maximum value, with enough iterations the sequence would have to hit a repeating pattern.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes an initial seed and returns the repeating pattern and the index of the sequence where the pattern begins. With an initial seed of 7, the sequence would be 7, 8, 4, 7, 6, 12, 4, 7, 6, 12,… Therefore, the repeating pattern is [7 6 12 4] and the start index is 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