Problem 52769. Easy Sequences 22: Sum of Proper Fractions

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>Let 'F' be the set of all proper fractions in lowest term, whose denominator is less than or equal 'd'. So, for d = 10, we have:</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>F(10)=\\left \\{ \\frac{1}{2},\\frac{1}{3},\\frac{1}{4},\\frac{1}{5},\\frac{1}{6},\\frac{1}{7},\\frac{1}{8},\\frac{1}{9},\\frac{1}{10},\\frac{2}{3},\\frac{2}{5},\\frac{2}{7},\\frac{2}{9},\\frac{3}{4},\\frac{3}{5},\\frac{3}{7},\\frac{3}{8},\\frac{3}{10},\\frac{4}{5},\\frac{4}{7},\\frac{4}{9},\\frac{5}{6},\\frac{5}{7},\\frac{5}{8},\\frac{5}{9},\\frac{6}{7},\\frac{7}{8},\\frac{7}{9},\\frac{7}{10},\\frac{8}{9},\\frac{9}{10}  \\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>Note that '2/4' and '3/9' are not included because they are not in their lowest term, while '7/7' and '8/5' are not included because they are not proper fractions.</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:rPr><w:b/></w:rPr><w:t>Write a function 'S(d)', which is the integer part of the sum of 'F(d)'.</w:t></w:r><w:r><w:t>  For the case above, the sum of elements of 'F(10)' is exactly '15.5'. Therefore S(10) = 15.</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: 188px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 94px; transform-origin: 407px 94px; 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: 368.5px 8px; transform-origin: 368.5px 8px; unicode-bidi: normal; "><span style="">Let 'F' be the set of all proper fractions in lowest term, whose denominator is less than or equal 'd'. So, for d = 10, we have:</span></span></div><div style="block-size: 35px; 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 17.5px; text-align: left; transform-origin: 384px 17.5px; 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: 24px 8px; transform-origin: 24px 8px; unicode-bidi: normal; "><span style="">            </span></span><span style="vertical-align:-15px"><img 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" style="width: 604px; height: 35px;" width="604" height="35"></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: 374px 8px; transform-origin: 374px 8px; unicode-bidi: normal; "><span style="">Note that '2/4' and '3/9' are not included because they are not in their lowest term, while '7/7' and '8/5' are not included because they are not proper fractions.</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: 233px 8px; transform-origin: 233px 8px; unicode-bidi: normal; "><span style="font-weight: 700; ">Write a function 'S(d)', which is the integer part of the sum of 'F(d)'.</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; perspective-origin: 143px 8px; transform-origin: 143px 8px; unicode-bidi: normal; "><span style="">  For the case above, the sum of elements of 'F(10)' is exactly '15.5'. Therefore S(10) = 15.</span></span></div></div></div>

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9 Solutions
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Last Solution submitted on Jan 05, 2022

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David Hill on 23 Sep 2021

Could not figure out how to speed up my code. Ended up having to use a lookup table to speed it up sufficiently.

Ramon Villamangca on 28 Sep 2021

Hi,

There is a shortcut using Mobius Inversion...

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Solution 6597389

2 Comments

2 Comments

Are Mjaavatten on 28 Sep 2021

I too had problems with the time limit for large arguments and had to resort to a few tricks. But my solution is still general.

Ramon Villamangca on 28 Sep 2021

There is a shortcut using Mobius Inversion...

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Problem 52769. Easy Sequences 22: Sum of Proper Fractions

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Problem 52769. Easy Sequences 22: Sum of Proper Fractions

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