Problem 52085. Compute a sum involving the totient over divisors of a number

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>Write a function to compute the following sum:</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=\"y = sum_(d|n) (-1)^(n/d) phi(d)\"/></w:customXmlPr><w:r><w:t>y = \\sum_{d|n}(-1)^{n/d}\\varphi(d)</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>where </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(d)\"/></w:customXmlPr><w:r><w:t>\\varphi(d)</w:t></w:r></w:customXml><w:r><w:t> is the </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/656\"><w:r><w:t>totient function</w:t></w:r></w:hyperlink><w:r><w:t>. The sum is computed over the divisors 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> (including 1 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=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t>). The input to the function will be two limits </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=\"a\"/></w:customXmlPr><w:r><w:t>a</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=\"b\"/></w:customXmlPr><w:r><w:t>b</w:t></w:r></w:customXml><w:r><w:t>. Compute </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>y</w:t></w:r></w:customXml><w:r><w:t> for </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=\"a &lt;= n &lt;= b\"/></w:customXmlPr><w:r><w:t>a \\le n \\le b</w:t></w:r></w:customXml><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: 116.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 58.2333px; transform-origin: 407px 58.2333px; vertical-align: baseline; "><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: 143.008px 7.79167px; transform-origin: 143.008px 7.79167px; unicode-bidi: normal; ">Write a function to compute the following sum:</div><div style="block-size: 35.4667px; 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 17.7333px; text-align: left; transform-origin: 384px 17.7333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><img src="data:image/png;base64,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" alt="y = sum_(d|n) (-1)^(n/d) phi(d)" style="width: 126px; height: 35.5px;" width="126" height="35.5"></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: 21.0083px 7.79167px; transform-origin: 21.0083px 7.79167px; unicode-bidi: normal; ">where <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEEAAAAlCAYAAAAdkoQJAAACtUlEQVRoge2ZXbHjMAyFPw5lEAIhEASLoAzKIAxKIRgKIRxCoRhK4e6DfdaqN3bsxN3pzuTM9OXGlmT9HMm+cOLEiRPfgQvQN5bZebn/BS7ATBsnXAgH7xvK/ShaOGAAfgGjl9WZbx3wPCj/41iA24H9V+AO/PjfsrKmxzniK0vjjotcC8gJ94yuRyNdzdDhjB4aysrJuwAvXNl8DSbaZcEV54AX+ZSfcGXxFVDkro3kPby8Laf2tMu+w7jhjOm2Fhbi5eWNhWunRnoZvFLbejrcAUfyUZ7Zn5adlz3i6nsg8EFJGzyi+w9uXsjCeyTVpkZCpJ+sR7skdWN0uAi+vK7B27AQ+KAEo1+/u11OXkDcaiTYpuOSWCs+SLWyNQy4Q754j7bNgtL2J1t38YIcEA8dOlQciZl1xpbhJfULLuUlJzbcOqGUZCWvulUqvdc2i5ljspnNnr1O6MmTnlpjDcnWBgEIQ8baSJobUrTnSCbIkWujMITsTH1fwy4nqIbWUk7f4oP2Zk9MgKVG2CinUv3JPn6pnlGkaG0aS5GfLZ/4ciQHbTkhxSmCaruW5KqJ0aZ7rhTig1rHpVrkFpunMknYclIKckLxoGbZN045G23btmwap67I6u979Ma65ST7mJLDg/KZAniv7dgYdQU7fV0IWZAbhjRYpZBzQk8oQ1tWE2V1/mTHlVosH2+UEbY1iq1n8lGRc3O92nYXZZqm1ZisJ8ruA9JbfXGz6S2jLSldcQe2I20JnuQNtymvn57J+ujvpTpHKkvBYiC8As28p6NebOSMUsi5WxkzEi5L8f4bdTdRZVETlNR9CWbqevwRqJSawLbGo17t+PtS9Ak015NqjXvxL16CF9q9YgGhNe4mmBUMbHeUvShtnVWovb+Xoqe9sXp5aoqOwNZf/R+dEydOnGiF36PeHdxmJU+TAAAAAElFTkSuQmCC" alt="phi(d)" style="width: 32.5px; height: 18.5px;" width="32.5" 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: 20.6083px 7.79167px; transform-origin: 20.6083px 7.79167px; unicode-bidi: normal; "> is 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: 133.4px 7.79167px; transform-origin: 133.4px 7.79167px; unicode-bidi: normal; ">. The sum is computed over the divisors of 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: 53.3px 7.79167px; transform-origin: 53.3px 7.79167px; unicode-bidi: normal; "> (including 1 and 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: 83.225px 7.79167px; transform-origin: 83.225px 7.79167px; unicode-bidi: normal; ">). The input to the function will be two limits 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 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; "> and b<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: 34.225px 7.79167px; transform-origin: 34.225px 7.79167px; unicode-bidi: normal; ">. Compute y<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: 12.05px 7.79167px; transform-origin: 12.05px 7.79167px; unicode-bidi: normal; "> for <img src="data:image/png;base64,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" alt="a <= n <= b" style="width: 62.5px; height: 18px;" width="62.5" height="18"><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 7.79167px; transform-origin: 3.88333px 7.79167px; unicode-bidi: normal; ">. </div></div></div>

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Problem 52085. Compute a sum involving the totient over divisors of a number

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Problem 52085. Compute a sum involving the totient over divisors of a number

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