Problem 53810. List the Beatriz 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>Write a function to find integers </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> less than or equal to an input value </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=\"m\"/></w:customXmlPr><w:r><w:t>m</w:t></w:r></w:customXml><w:r><w:t> that solve the equation </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-1) = phi(n+1)\"/></w:customXmlPr><w:r><w:t>\\sigma(n-1) = \\phi(n+1)</w:t></w:r></w:customXml><w:r><w:t>. The function </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(x)\"/></w:customXmlPr><w:r><w:t>\\sigma(x)</w:t></w:r></w:customXml><w:r><w:t> gives the </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/46898\"><w:r><w:rPr><w:u/></w:rPr><w:t>sum of divisors</w:t></w:r></w:hyperlink><w:r><w:t> 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=\"x\"/></w:customXmlPr><w:r><w:t>x</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=\"sigma(15) = 1+3+5+15 = 24\"/></w:customXmlPr><w:r><w:t>\\sigma(15) = 1+3+5+15 = 24</w:t></w:r></w:customXml><w:r><w:t>. The </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>totient function</w:t></w:r></w:hyperlink><w:r><w:t> </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(x)\"/></w:customXmlPr><w:r><w:t>\\phi(x)</w:t></w:r></w:customXml><w:r><w:t> counts 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=\"x\"/></w:customXmlPr><w:r><w:t>x</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=\"x\"/></w:customXmlPr><w:r><w:t>x</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(15) = 8\"/></w:customXmlPr><w:r><w:t>\\phi(15) = 8</w:t></w:r></w:customXml><w:r><w:t> because 1, 2, 4, 7, 8, 11, 13, and 14 are relatively prime to 15; that is, the greatest common divisor of 15 and each of the eight numbers 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>An oblique hint is that I call solutions 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=\"sigma(n-1) = phi(n+1)\"/></w:customXmlPr><w:r><w:t>\\sigma(n-1) = \\phi(n+1)</w:t></w:r></w:customXml><w:r><w:t> “Beatriz numbers” in honor of a good friend and former co-worker of </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/players/18927291\"><w:r><w:t>EmilyR</w:t></w:r></w:hyperlink><w:r><w:t> and </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/players/8608872\"><w:r><w:t>JessicaR</w:t></w:r></w:hyperlink><w:r><w:t>. The natural search will not help, unless you are interested in combinatorics related to Higham’s conjecture.</w:t></w:r><w:r><w:t> </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: 156px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407.5px 78px; transform-origin: 407.5px 78px; 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: 384.5px 42px; text-align: left; transform-origin: 384.5px 42px; 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: 98.275px 7.66667px; transform-origin: 98.275px 7.66667px; unicode-bidi: normal; ">Write a function to find integers n<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: 112.417px 7.66667px; transform-origin: 112.417px 7.66667px; unicode-bidi: normal; "> less than or equal to an input value m<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: 74.2917px 7.66667px; transform-origin: 74.2917px 7.66667px; unicode-bidi: normal; "> that solve the equation <img src="data:image/png;base64,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" alt="sigma(n-1) = phi(n+1)" style="width: 127px; height: 18.5px;" width="127" height="18.5"><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: 17.8833px 7.66667px; transform-origin: 17.8833px 7.66667px; unicode-bidi: normal; ">. The function <img src="data:image/png;base64,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" alt="sigma(x)" style="width: 30.5px; height: 18.5px;" width="30.5" height="18.5"><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: 31.8917px 7.66667px; transform-origin: 31.8917px 7.66667px; unicode-bidi: normal; "> gives the <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/46898">sum of divisors</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: 9.71667px 7.66667px; transform-origin: 9.71667px 7.66667px; unicode-bidi: normal; "> of x<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: 44.3333px 7.66667px; transform-origin: 44.3333px 7.66667px; unicode-bidi: normal; ">; for example, <img src="data:image/png;base64,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" alt="sigma(15) = 1+3+5+15 = 24" style="width: 178.5px; height: 18.5px;" width="178.5" height="18.5"><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: 17.8833px 7.66667px; transform-origin: 17.8833px 7.66667px; unicode-bidi: normal; ">. The <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/656">totient function</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: 1.94167px 7.66667px; transform-origin: 1.94167px 7.66667px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" alt="phi(x)" style="width: 32px; height: 18.5px;" width="32" height="18.5"><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: 24.5px 7.66667px; transform-origin: 24.5px 7.66667px; unicode-bidi: normal; "> counts numbers less than x<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 7.66667px; transform-origin: 84.4px 7.66667px; unicode-bidi: normal; "> that are relatively prime to x<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: 46.6667px 7.66667px; transform-origin: 46.6667px 7.66667px; unicode-bidi: normal; ">. For example, <img src="data:image/png;base64,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" alt="phi(15) = 8" style="width: 66px; height: 18.5px;" width="66" height="18.5"><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: 127.433px 7.66667px; transform-origin: 127.433px 7.66667px; unicode-bidi: normal; "> because 1, 2, 4, 7, 8, 11, 13, and 14 are relatively prime to 15; that is, the greatest common divisor of 15 and each of the eight numbers is 1. </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: 384.5px 31.5px; text-align: left; transform-origin: 384.5px 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: 122.525px 7.66667px; transform-origin: 122.525px 7.66667px; unicode-bidi: normal; ">An oblique hint is that I call solutions to <img src="data:image/png;base64,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" alt="sigma(n-1) = phi(n+1)" style="width: 127px; height: 18.5px;" width="127" height="18.5"><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: 182.817px 7.66667px; transform-origin: 182.817px 7.66667px; unicode-bidi: normal; "> “Beatriz numbers” in honor of a good friend and former co-worker of <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/players/18927291">EmilyR</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: 15.5583px 7.66667px; transform-origin: 15.5583px 7.66667px; unicode-bidi: normal; "> and <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/players/8608872">JessicaR</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: 268.4px 7.66667px; transform-origin: 268.4px 7.66667px; unicode-bidi: normal; ">. The natural search will not help, unless you are interested in combinatorics related to Higham’s conjecture.<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 7.66667px; transform-origin: 3.88333px 7.66667px; unicode-bidi: normal; "> </div></div></div>

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Problem 53810. List the Beatriz numbers

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