Problem 3003. Mobius function

Created by goc3

Solve Later

{"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","targetMode":"","relationshipId":"rId1","target":"/matlab/document.xml"},{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/output","targetMode":"","relationshipId":"rId2","target":"/matlab/output.xml"}],"parts":[{"partUri":"/matlab/document.xml","relationship":[],"contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>From</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://en.wikipedia.org/wiki/Möbius_function\"><w:r><w:t>wikipedia</w:t></w:r></w:hyperlink><w:r><w:t>:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>For any positive integer n, define μ(n) as the sum of the primitive n-th roots of unity. It has values in {−1, 0, 1} depending on the factorization of n into prime factors:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>μ(n) = 1 if n is a square-free positive integer with an even number of prime factors.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>μ(n) = −1 if n is a square-free positive integer with an odd number of prime factors.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>μ(n) = 0 if n has a squared prime factor.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Return numbers from the</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://oeis.org/A008683\"><w:r><w:t>Mobius function sequence</w:t></w:r></w:hyperlink><w:r><w:t> corresponding to the supplied indices. For example, if n = 3:7, your function should return [-1, 0, -1, 1, -1].</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Hint: solving</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/3001-sphenic-number-sequence\"><w:r><w:t>Problem 3001</w:t></w:r></w:hyperlink><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/3002-not-square-free-number-sequence\"><w:r><w:t>Problem 3002</w:t></w:r></w:hyperlink><w:r><w:t> will provide much of the code needed for this problem. You'll need to add prime numbers to the sphenic number set (resulting from Problem 3001).</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

From <a href = "http://en.wikipedia.org/wiki/Möbius_function">wikipedia</a>:For any positive integer n, define μ(n) as the sum of the primitive n-th roots of unity. It has values in {−1, 0, 1} depending on the factorization of n into prime factors:<ul><li>μ(n) = 1 if n is a square-free positive integer with an even number of prime factors.</li><li>μ(n) = −1 if n is a square-free positive integer with an odd number of prime factors.</li><li>μ(n) = 0 if n has a squared prime factor.</li></ul>Return numbers from the <a href = "http://oeis.org/A008683">Mobius function sequence</a> corresponding to the supplied indices. For example, if n = 3:7, your function should return [-1, 0, -1, 1, -1].Hint: solving <a href = "https://www.mathworks.com/matlabcentral/cody/problems/3001-sphenic-number-sequence">Problem 3001</a> and <a href = "https://www.mathworks.com/matlabcentral/cody/problems/3002-not-square-free-number-sequence">Problem 3002</a> will provide much of the code needed for this problem. You'll need to add prime numbers to the sphenic number set (resulting from Problem 3001).

From <http://en.wikipedia.org/wiki/Möbius_function wikipedia>:

For any positive integer n, define μ(n) as the sum of the primitive n-th roots of unity. It has values in {−1, 0, 1} depending on the factorization of n into prime factors:

* μ(n) = 1 if n is a square-free positive integer with an even number of prime factors.
* μ(n) = −1 if n is a square-free positive integer with an odd number of prime factors.
* μ(n) = 0 if n has a squared prime factor.

Return numbers from the <http://oeis.org/A008683 Mobius function sequence> corresponding to the supplied indices. For example, if n = 3:7, your function should return [-1, 0, -1, 1, -1].

Hint: solving <https://www.mathworks.com/matlabcentral/cody/problems/3001-sphenic-number-sequence Problem 3001> and <https://www.mathworks.com/matlabcentral/cody/problems/3002-not-square-free-number-sequence Problem 3002> will provide much of the code needed for this problem. You'll need to add prime numbers to the sphenic number set (resulting from Problem 3001).

Solve

Solution Stats

83 Solutions
39 Solvers

Last Solution submitted on Nov 11, 2021

Last 200 Solutions

Problem Comments

3 Comments

3 Comments

Jan Orwat on 11 Feb 2015

Hello Grant. It's a nice set, but for problems 3001 and 3002 lookup tables are "always win" solutions. For this problem you can avoid this by using some randomness in the test suite.

goc3 on 11 Feb 2015

I'm glad you like them, and thanks for the feedback. I just added a random test case to all three problems.

rifat on 11 Feb 2015

Yeah, the series is good. After a few rounds of spam-storm, Cody seems to be back on track...

Problem Recent Solvers39

Problem Tags

factor integer mobius number prime sequence sphenic square-free

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!