Problem 52689. Easy Sequences 18: Set Bits of Triple Summations

Created by Ramon Villamangca

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>The function S(n) is defined by the following triple summations:</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>                            </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>S(n)=\\left [\\sum_{k=2}^{n}\\sum_{j=2}^{k} \\sum_{i=2}^{j}\\frac{1}{\\log {_{i}}{\\left ( j! \\right )}}  \\right ]</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>The double brackets mean that the output of the triple summations is being rounded-off to the nearest integer. </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>Write the function 'bitS(n)', which is the number of </w:t></w:r><w:hyperlink w:docLocation=\"https://www.quora.com/What-do-we-mean-by-a-set-bit\"><w:r><w:t>bits set</w:t></w:r></w:hyperlink><w:r><w:rPr><w:b/></w:rPr><w:t> in the binary representation of S(n).</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: 20.440000534057617px; 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: normal; text-decoration: none; white-space: normal; "><div style="block-size: 127px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; 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; text-align: left; 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; "><span style="">The function S(n) is defined by the following triple summations:</span></span></div><div style="block-size: 46px; 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; text-align: left; 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; "><span style="">                            </span></span><span style="vertical-align:-17px"><img 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" width="186" height="46" style="width: 186px; height: 46px;"></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; text-align: left; 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; "><span style="">The double brackets mean that the output of the triple summations is being rounded-off to the nearest integer. </span></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; "><span style="font-weight: bold; ">Write the function 'bitS(n)', which is the number of </span></span><a target='_blank' href = "https://www.quora.com/What-do-we-mean-by-a-set-bit"><span style=""><span style="">bits set</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; "><span style="font-weight: bold; "> in the binary representation of S(n).</span></span></div></div></div>

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Last Solution submitted on Dec 28, 2021

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William on 12 Sep 2021

Hmmm. It's a very clever problem, but I am getting different answers!

ChrisR on 12 Sep 2021

I am as well. For example, for n = 100, I get 4 (out of 7) bits set.

William on 12 Sep 2021

Chris -- Yes, exactly.

Ramon Villamangca on 12 Sep 2021

Let me check. Thanks.

Ramon Villamangca on 12 Sep 2021

Hi Chris/William,

Should be 'triple summations', sorry for that.

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binary easy sequences

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Problem 52689. Easy Sequences 18: Set Bits of Triple Summations

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Problem 52689. Easy Sequences 18: Set Bits of Triple Summations

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