Problem 51715. Iterate the sum of divisors and totient

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:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/46898\"><w:r><w:rPr><w:u/></w:rPr><w:t>Cody Problem 46898</w:t></w:r></w:hyperlink><w:r><w:t> deals with the sum of divisors function, denoted by </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=\"sigma(n)\"/></w:customXmlPr><w:r><w:t>\\sigma(n)</w:t></w:r></w:customXml><w:r><w:t>, while </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/656\"><w:r><w:rPr><w:u/></w:rPr><w:t>Cody Problem 656</w:t></w:r></w:hyperlink><w:r><w:t> deals with the totient function, denoted by </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=\"phi(n)\"/></w:customXmlPr><w:r><w:t>\\varphi(n)</w:t></w:r></w:customXml><w:r><w:t>. The sum of divisors is straightforward: for example, </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=\"sigma(12) = 1+2+3+4+6+12 = 28\"/></w:customXmlPr><w:r><w:t>\\sigma(12) = 1+2+3+4+6+12 = 28</w:t></w:r></w:customXml><w:r><w:t>. The totient of </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=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t> counts the numbers less than </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=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t> that are relatively prime to </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=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t>. For example </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=\"phi(12) = 4\"/></w:customXmlPr><w:r><w:t>\\varphi(12) = 4</w:t></w:r></w:customXml><w:r><w:t> because the greatest common divisor of 12 and four numbers (1, 5, 7, 11) is 1. </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>What happens if you repeatedly apply the two functions, starting with the sum of divisors and alternating? For example, start with 7. Then </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"sigma(7) = 8, phi(8) = 4, sigma(4) = 7, phi(7) = 6, sigma(6) = 12, phi(12) = 4, \"/></w:customXmlPr><w:r><w:t>\\sigma(7) = 8, \\varphi(8) = 4, \\sigma(4) = 7, \\varphi(7) = 6, \\sigma(6) = 12, \\varphi(12) = 4,</w:t></w:r></w:customXml><w:r><w:t> etc.</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>and the pattern 7, 6, 12, 4 will repeat. </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>Oscillating behavior is plausible because the sum of divisors is always greater than </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=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t>, and the totient is always smaller than </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=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t>. 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.</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 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.</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: 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; "><div 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; "><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/46898"><span style=""><span style="text-decoration: underline; text-decoration-line: underline; ">Cody Problem 46898</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: 160.25px 7.91667px; transform-origin: 160.25px 7.91667px; unicode-bidi: normal; "><span style=""> deals with the sum of divisors function, denoted by </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="sigma(n)" style="width: 30.5px; height: 19px;" width="30.5" height="19"></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: 21.7833px 7.91667px; transform-origin: 21.7833px 7.91667px; unicode-bidi: normal; "><span style="">, while </span></span><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/656"><span style=""><span style="text-decoration: underline; text-decoration-line: underline; ">Cody Problem 656</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: 46.675px 7.91667px; transform-origin: 46.675px 7.91667px; unicode-bidi: normal; "><span style=""> deals with the totient function, denoted by </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="phi(n)" style="width: 31.5px; height: 19px;" width="31.5" height="19"></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: 164.125px 7.91667px; transform-origin: 164.125px 7.91667px; unicode-bidi: normal; "><span style="">. The sum of divisors is straightforward: for example, </span></span><span style="vertical-align:-5px"><img 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"></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: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; "><span style="">. The totient of </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</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: 95.2917px 7.91667px; transform-origin: 95.2917px 7.91667px; unicode-bidi: normal; "><span style=""> counts the numbers less than </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</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: 84.4px 7.91667px; transform-origin: 84.4px 7.91667px; unicode-bidi: normal; "><span style=""> that are relatively prime to </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</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: 44.725px 7.91667px; transform-origin: 44.725px 7.91667px; unicode-bidi: normal; "><span style="">. For example </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="phi(12) = 4" style="width: 65px; height: 19px;" width="65" height="19"></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: 68.5083px 7.91667px; transform-origin: 68.5083px 7.91667px; unicode-bidi: normal; "><span style=""> because the greatest common divisor of 12 and four numbers (1, 5, 7, 11) is 1. </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: 384px 7.91667px; transform-origin: 384px 7.91667px; unicode-bidi: normal; "><span style="">What happens if you repeatedly apply the two functions, starting with the sum of divisors and alternating? For example, start with 7. Then </span></span></div><div 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; "><span style="vertical-align:-5px"><img 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"></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: 13.2167px 7.91667px; transform-origin: 13.2167px 7.91667px; unicode-bidi: normal; "><span style=""> etc.</span></span></div><div 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; "><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: 117.458px 7.91667px; transform-origin: 117.458px 7.91667px; unicode-bidi: normal; "><span style="">and the pattern 7, 6, 12, 4 will repeat. </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: 258.3px 7.91667px; transform-origin: 258.3px 7.91667px; unicode-bidi: normal; "><span style="">Oscillating behavior is plausible because the sum of divisors is always greater than </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</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: 120.45px 7.91667px; transform-origin: 120.45px 7.91667px; unicode-bidi: normal; "><span style="">, and the totient is always smaller than </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</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: 360.958px 7.91667px; transform-origin: 360.958px 7.91667px; unicode-bidi: normal; "><span style="">. 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.</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: 383.4px 7.91667px; transform-origin: 383.4px 7.91667px; unicode-bidi: normal; "><span style="">Write 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.</span></span></div></div></div>

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David Hill on 10 May 2021

Chris,
Would you check test17. I get a repeating pattern starting at index 161 that has 36 members.

ChrisR on 10 May 2021

Yes, I do too. Not sure what happened or why my reference solution worked. Thanks, David.

goc3 on 7 May 2022

An interesting and challenging problem. I feel that it should be worth more than ten points...

Rafael S.T. Vieira on 27 Nov 2022

The greatest challenge is to produce the most optimized code. Writing code that works is easy.

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Problem 51715. Iterate the sum of divisors and totient

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Problem 51715. Iterate the sum of divisors and totient

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