Problem 47843. Compute the arithmetic derivative of integers

Created by ChrisR

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{"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>The Lagarias </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Arithmetic_derivative\"><w:r><w:t>arithmetic derivative</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=\"D(n)\"/></w:customXmlPr><w:r><w:t>D(n)</w:t></w:r></w:customXml><w:r><w:t> is defined for integers such that the derivative of a prime number is 1: </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=\"D(p) = 1\"/></w:customXmlPr><w:r><w:t>D(p) = 1</w:t></w:r></w:customXml><w:r><w:t>. It also satisfies the Leibniz formula, which resembles the product rule of calculus:</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=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"D(mn) = mD(n)+nD(m)\"/></w:customXmlPr><w:r><w:t>D(mn) = mD(n)+nD(m)</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>For example, the arithmetic derivative of 15 is </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=\"3D(5) + 5D(3) = 3x1 + 5x1 = 8\"/></w:customXmlPr><w:r><w:t>3D(5) + 5D(3) = 3\\cdot1 + 5\\cdot1 = 8</w:t></w:r></w:customXml><w:r><w:t>. The Leibniz formula also implies that </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=\"D(1) = 0\"/></w:customXmlPr><w:r><w:t>D(1) = 0</w:t></w:r></w:customXml><w:r><w:t>. </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 compute the arithmetic derivative of a natural number. </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: 153px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 76.5px; transform-origin: 407px 76.5px; 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: 42.7917px 7.91667px; transform-origin: 42.7917px 7.91667px; unicode-bidi: normal; "><span style="">The Lagarias </span></span><a target='_blank' href = "https://en.wikipedia.org/wiki/Arithmetic_derivative"><span style=""><span style="">arithmetic derivative</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: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; "><span style=""> </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="D(n)" style="width: 34px; height: 19px;" width="34" 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: 218.2px 7.91667px; transform-origin: 218.2px 7.91667px; unicode-bidi: normal; "><span style=""> is defined for integers such that the derivative of a prime number is 1: </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="D(p) = 1" style="width: 61.5px; height: 19px;" width="61.5" 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: 9.70833px 7.91667px; transform-origin: 9.70833px 7.91667px; unicode-bidi: normal; "><span style="">. It also satisfies the Leibniz formula, which resembles the product rule of calculus:</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="D(mn) = mD(n)+nD(m)" style="width: 162px; height: 19px;" width="162" height="19"></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: 143.133px 7.91667px; transform-origin: 143.133px 7.91667px; unicode-bidi: normal; "><span style="">For example, the arithmetic derivative of 15 is </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="3D(5) + 5D(3) = 3x1 + 5x1 = 8" style="width: 210px; height: 19px;" width="210" 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: 119.417px 7.91667px; transform-origin: 119.417px 7.91667px; unicode-bidi: normal; "><span style="">. The Leibniz formula also implies that </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHgAAAAmCAYAAAAP4F9VAAADO0lEQVR4nO2abZGDMBCGXw84wAAGqqAKcICDOqiF04AEPNQCGmqh94PssGSAbDYpNMw+M0znbiBZ8mY/kgAYhmEYhmEYl6YC8Od+c3AH0GVq69LcADyE103ZRwVgANBE3C8R74Fp0lyFBtN7P9xvnath8q4Pu0bMwj4BvNz/3+5vaeeVe1YyOSrX39tdEv6cPSXTYHKAN6bIRE73wfRuWaLeDUuB17zt7owgoSWi9ZAJ0LG2qX0po7OtRBrM7+07DYk8IIPI1Bh5755BXIQ9kVt3T8i4zrVTub5jBb4J+/lFKDJuOQGNR3K9MWAWLpTXeDh/7dxHYV5jR4zAcHaUlo9bzOO45ShPzOORlJN5eA6FO+7FW8aR8bFepRVY29+Z8Lpmy+475nFutR3xRqSDxO9fCy8vTGLFohW4QqZQdhA1ZFGQ39drO6MwQAldAg/p/jNkVGx45u3GCgzoJ9UZ8KI2JJym8FxAoSJGFC6wPwMpXGrWzSkCU22goYV8P2DvksKLWqnAa5V2EAptoWS/16kvMBkv3djgpAico9+UK8ZmLnCoOOTLx+h345Wc1EB/UvgG0mBpSBG4gz5y5PDgmPzPBQ49xydftMB8ySNN4n5R5ht4lsCU10ootLjAoY2gEQkC84elA+Nva/p54WyBNcXd0VC0+WoO5iW49OEKy5ywZpwJHOaQKprPor3tSQ4PLW+sh4yzBE6p3o+uonkds7e0k963Ss8elhwI1Fh671ZIp3W1ZlcpRxWdsjw7qooGljtZW3BPj45MXKzQ9iQd+0nCSko1myIwTSwNR1fR1Gdoeco3oaIcxj8e3Hu4RtxmCO1Va6rZFIEH7G/7/SLS06Ro7+UzY2uzmx/AU56WnruO0J3uSMLWGpSrSjv8p/PgtXqGImH0eTCd03IP7rEMN/5ec+xJxhNxItVYFnA0a6UvRmvzbJ+5HEiDySFGTOPMv+jooRBXkk+oIy20BJNMjNB3YZKo0aOcg4Yt+Di0KGCyPnHMoNNk0uxBGwnQJzjqg2ohA8rLvZeB8su3wk2H8irny3FHpi8DPRpM4pb0mc5loW9/c4nRQFFhGt8lpxg/X2EahmEYhmEYhfIPnUvDeXKmK2EAAAAASUVORK5CYII=" alt="D(1) = 0" style="width: 60px; height: 19px;" width="60" 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="">. </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: 227.017px 7.91667px; transform-origin: 227.017px 7.91667px; unicode-bidi: normal; "><span style="">Write a function to compute the arithmetic derivative of a natural number. </span></span></div></div></div>

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Problem 47843. Compute the arithmetic derivative of integers

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Problem 47843. Compute the arithmetic derivative of integers

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