Problem 50913. Compute the nth term from the golden sieve

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:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/45367\"><w:r><w:rPr><w:u/></w:rPr><w:t>Cody Problem 45367</w:t></w:r></w:hyperlink><w:r><w:t> involved the Sieve of Eratosthenes, and </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/50811-compute-the-nth-term-from-the-sieve-of-flavius-josephus\"><w:r><w:rPr><w:u/></w:rPr><w:t>Cody Problem 50811</w:t></w:r></w:hyperlink><w:r><w:t> involved the Sieve of Flavius Josephus. To apply the </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>golden sieve</w:t></w:r><w:r><w:t>, start with the natural numbers and at the </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>th step, take the sequence, which we will call </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 delete the term in the </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(n)\"/></w:customXmlPr><w:r><w:t>a(n)</w:t></w:r></w:customXml><w:r><w:t> position. </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>In the first step, delete the term in the </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>a(1) = 1</w:t></w:r></w:customXml><w:r><w:t> position to get 2, 3, 4, 5, 6, 7,…Then in the second step, delete the term in the </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(2) = 3\"/></w:customXmlPr><w:r><w:t>a(2) = 3</w:t></w:r></w:customXml><w:r><w:t> position (i.e., 4) to get 2, 3, 5, 6, 7, 8, 9,…In the third step, delete the term in the </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(3) = 5\"/></w:customXmlPr><w:r><w:t>a(3) = 5</w:t></w:r></w:customXml><w:r><w:t> position (i.e., 7) to get 2, 3, 5, 6, 8, 9, 10,…Et cetera.</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 that returns the </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>th term in the sequence after an infinite number of steps.</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: 165px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 82.5px; transform-origin: 407px 82.5px; vertical-align: baseline; "><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: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/45367">Cody Problem 45367</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: 129.5px 8px; transform-origin: 129.5px 8px; unicode-bidi: normal; "> involved the Sieve of Eratosthenes, and <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/50811-compute-the-nth-term-from-the-sieve-of-flavius-josephus">Cody Problem 50811</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: 94px 8px; transform-origin: 94px 8px; unicode-bidi: normal; "> involved the Sieve of Flavius Josephus. To apply the <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: 39.5px 8px; transform-origin: 39.5px 8px; unicode-bidi: normal; ">golden sieve<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: 133.5px 8px; transform-origin: 133.5px 8px; unicode-bidi: normal; ">, start with the natural numbers and at the 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: 129px 8px; transform-origin: 129px 8px; unicode-bidi: normal; ">th step, take the sequence, which we will call 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: 87px 8px; transform-origin: 87px 8px; unicode-bidi: normal; ">, and delete the term in the <img src="data:image/png;base64,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" alt="a(n)" style="width: 29.5px; height: 18.5px;" width="29.5" 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: 30.5px 8px; transform-origin: 30.5px 8px; unicode-bidi: normal; "> position. </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: 384px 31.5px; text-align: left; transform-origin: 384px 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: 119.5px 8px; transform-origin: 119.5px 8px; unicode-bidi: normal; ">In the first step, delete the term in the <img src="data:image/png;base64,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" style="width: 56px; height: 18.5px;" width="56" 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: 231.5px 8px; transform-origin: 231.5px 8px; unicode-bidi: normal; "> position to get 2, 3, 4, 5, 6, 7,…Then in the second step, delete the term in the <img src="data:image/png;base64,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" alt="a(2) = 3" style="width: 56px; height: 18.5px;" width="56" 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: 256px 8px; transform-origin: 256px 8px; unicode-bidi: normal; "> position (i.e., 4) to get 2, 3, 5, 6, 7, 8, 9,…In the third step, delete the term in the <img src="data:image/png;base64,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" alt="a(3) = 5" style="width: 56px; height: 18.5px;" width="56" 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: 52.7167px 8px; transform-origin: 52.7167px 8px; unicode-bidi: normal; "> position (i.e., 7) to get 2, 3, 5, 6, 8, 9, 10,…Et cetera.</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: 102px 8px; transform-origin: 102px 8px; unicode-bidi: normal; ">Write a function that returns the 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: 178.5px 8px; transform-origin: 178.5px 8px; unicode-bidi: normal; ">th term in the sequence after an infinite number of steps.</div></div></div>

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Last Solution submitted on Dec 04, 2022

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ChrisR on 12 Mar 2021

OK, but you can you *derive* that formula?

Tim on 12 Mar 2021

Does “look up in OEIS” count as “derive”?

ChrisR on 13 Mar 2021

Tim: You, William, David, and others have figured me out. I suppose there are worse habits than repeatedly clicking the "Webcam" button on OEIS.

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Problem 50913. Compute the nth term from the golden sieve

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Problem 50913. Compute the nth term from the golden sieve

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