Problem 52060. Identify Ruth-Aaron numbers

Created by ChrisR

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>After </w:t></w:r><w:hyperlink w:docLocation=\"https://www.youtube.com/watch?v=QjqYThEVoSQ\"><w:r><w:rPr><w:u/></w:rPr><w:t>Hank Aaron hit his 715</w:t></w:r><w:r><w:rPr><w:u/></w:rPr><w:t>th</w:t></w:r><w:r><w:rPr><w:u/></w:rPr><w:t> home run</w:t></w:r></w:hyperlink><w:r><w:t> and passed Babe Ruth’s total of 714, Carl Pomerance noticed that the union of the prime factors of the numbers (</w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"714 = 2*3*7*17\"/></w:customXmlPr><w:r><w:t>714 = 2\\cdot 3\\cdot 7 \\cdot 17</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"715 = 5*11*13\"/></w:customXmlPr><w:r><w:t>715 = 5\\cdot 11 \\cdot 13</w:t></w:r></w:customXml><w:r><w:t>) comprises all of the prime numbers up to 17. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>He told a colleague that he found an interesting property of the two numbers, and when the colleague posed the problem to a class, a student observed that the sums of the prime factors of these two consecutive numbers were equal—in this case, 29. Prof. Pomerance and colleagues named numbers with this property “Ruth-Aaron numbers.”*</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Write a function to classify numbers in this way. In computing sums, include only the unique factors. If a number is the lesser of a Ruth-Aaron pair, return ‘R’. If it is the greater number, return ‘A’. If multiple pairs overlap, return ‘RA’ for numbers that can serve as either the lesser or greater. If the number is not part of a Ruth-Aaron pair, return ‘X’.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>*The story continues in a </w:t></w:r><w:hyperlink w:docLocation=\"https://www.youtube.com/watch?v=aCq04N9it8U\"><w:r><w:rPr><w:u/></w:rPr><w:t>Numberphile video</w:t></w:r></w:hyperlink><w:r><w:t> on this topic, and I highly recommend that you watch it. Go for the number theory and stay for the life lessons!</w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div 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; "><div style="block-size: 237px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 118.5px; transform-origin: 407px 118.5px; vertical-align: baseline; "><div 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; "><span 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: 16.7167px 7.91667px; transform-origin: 16.7167px 7.91667px; unicode-bidi: normal; "><span style="">After </span></span><a target='_blank' href = "https://www.youtube.com/watch?v=QjqYThEVoSQ"><span style="perspective-origin: 70.8px 7.91667px; transform-origin: 70.8px 7.91667px; "><span style="text-decoration: underline; text-decoration-line: underline; ">Hank Aaron hit his 715</span></span><span style="perspective-origin: 5.83333px 7.91667px; transform-origin: 5.83333px 7.91667px; "><span style="text-decoration: underline; text-decoration-line: underline; ">th</span></span><span style="perspective-origin: 31.5083px 7.91667px; transform-origin: 31.5083px 7.91667px; "><span style="text-decoration: underline; text-decoration-line: underline; "> home run</span></span></a><span 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: 259.142px 7.91667px; transform-origin: 259.142px 7.91667px; unicode-bidi: normal; "><span style=""> and passed Babe Ruth’s total of 714, Carl Pomerance noticed that the union of the prime factors of the numbers (</span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="714 = 2*3*7*17" style="width: 114.5px; height: 18px;" width="114.5" height="18"></span><span 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: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; "><span style=""> and </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="715 = 5*11*13" style="width: 103.5px; height: 18px;" width="103.5" height="18"></span><span 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: 144.3px 7.91667px; transform-origin: 144.3px 7.91667px; unicode-bidi: normal; "><span style="">) comprises all of the prime numbers up to 17. </span></span></div><div 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; "><span 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: 383.158px 7.91667px; transform-origin: 383.158px 7.91667px; unicode-bidi: normal; "><span style="">He told a colleague that he found an interesting property of the two numbers, and when the colleague posed the problem to a class, a student observed that the sums of the prime factors of these two consecutive numbers were equal—in this case, 29. Prof. Pomerance and colleagues named numbers with this property “Ruth-Aaron numbers.”*</span></span></div><div 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; "><span 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: 365.35px 7.91667px; transform-origin: 365.35px 7.91667px; unicode-bidi: normal; "><span style="">Write a function to classify numbers in this way. In computing sums, include only the unique factors. If a number is the lesser of a Ruth-Aaron pair, return ‘R’. If it is the greater number, return ‘A’. If multiple pairs overlap, return ‘RA’ for numbers that can serve as either the lesser or greater. If the number is not part of a Ruth-Aaron pair, return ‘X’.</span></span></div><div 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; "><span 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: 78.9583px 7.91667px; transform-origin: 78.9583px 7.91667px; unicode-bidi: normal; "><span style="">*The story continues in a </span></span><a target='_blank' href = "https://www.youtube.com/watch?v=aCq04N9it8U"><span style=""><span style="text-decoration: underline; text-decoration-line: underline; ">Numberphile video</span></span></a><span 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: 232.967px 7.91667px; transform-origin: 232.967px 7.91667px; unicode-bidi: normal; "><span style=""> on this topic, and I highly recommend that you watch it. Go for the number theory and stay for the life lessons!</span></span></div></div></div>

After Hank Aaron hit his 715th home run and passed Babe Ruth’s total of 714, Carl Pomerance noticed that the union of the prime factors of the numbers ( and ) comprises all of the prime numbers up to 17. 
He told a colleague that he found an interesting property of the two numbers, and when the colleague posed the problem to a class, a student observed that the sums of the prime factors of these two consecutive numbers were equal—in this case, 29. Prof. Pomerance and colleagues named numbers with this property “Ruth-Aaron numbers.”*
Write a function to classify numbers in this way. In computing sums, include only the unique factors. If a number is the lesser of a Ruth-Aaron pair, return ‘R’. If it is the greater number, return ‘A’. If multiple pairs overlap, return ‘RA’ for numbers that can serve as either the lesser or greater. If the number is not part of a Ruth-Aaron pair, return ‘X’.
*The story continues in a Numberphile video on this topic, and I highly recommend that you watch it. Go for the number theory and stay for the life lessons!

Solve

Solution Stats

13 Solutions
9 Solvers

Last Solution submitted on Aug 21, 2021

Last 200 Solutions

Problem Comments

Problem Recent Solvers9

Problem Tags

baseball be flexible factor sum have fun number theory step up

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Problem 52060. Identify Ruth-Aaron numbers

Solution Stats

Last 200 Solutions

Problem Comments

Problem Recent Solvers9

Suggested Problems

More from this Author131

Problem Tags

Community Treasure Hunt

Problem 52060. Identify Ruth-Aaron numbers

Solution Stats

Last 200 Solutions

Problem Comments

Problem Recent Solvers9

Suggested Problems

More from this Author131

Problem Tags

Community Treasure Hunt

Players