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Search: id:A013929
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| A013929 |
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Numbers that are not square-free. Numbers that are divisible by a square greater than 1. The complement of A005117. |
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115
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| 4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Sometimes misnamed squareful numbers, but officially those are given by A001694.
This is different from the sequence of numbers n such that A007913(n)<phi(n). The two sequences differ at these values: 420,660,780,840,1320,1560,4620,5460,7140,..., which is essentially A070237. - Ant King (mathstutoring(AT)ntlworld.com), Dec 16 2005
Numbers n such that sum(d|n, d/phi(d)*mu(n/d))=0 - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 28 2002
Also n such there is at least one x<n such that A007913(x)=A007913(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 28 2002
Numbers for which there exists a partition into two parts p and q such that p+q = n and pq is a multiple of n. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 30 2003
Numbers n such there is a solution 0<x<n to x^2==0 mod(n). - Franz Vrabec (franz.vrabec(AT)planetuniqa.at), Aug 13 2005
Numbers n such that moebius(n) = 0.
a(n)=Such k that EulerPhi[k]/k = EulerPhi[m]/m for some m < k [From Artur Jasinski (grafix(AT)csl.pl), Nov 05 2008]
Appears to be numbers such that when a column with index equal to a(n) in A051731 is deleted, there is no impact on the result in the first column of A054525. [From Mats Granvik (mats.granvik(AT)abo.fi), Feb 06 2009]
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
S. Ramanujan, Irregular numbers, J. Indian Math. Soc. 5 (1913) 105-106.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Moebius Function
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MAPLE
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a := n -> `if`(numtheory[mobius](n)=0, n, NULL); seq(a(i), i=1..160); [From Peter Luschny (peter(AT)luschny.de), May 04 2009]
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MATHEMATICA
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Union[ Flatten[ Table[ n i^2, {i, 2, 20}, {n, 1, 400/i^2} ] ] ]
Select[ Range[2, 160], (Union[Last /@ FactorInteger[ # ]][[ -1]] > 1) == True &] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 11 2005)
a = {}; b = {}; Do[If[MemberQ[a, EulerPhi[n]/n], AppendTo[b, n], AppendTo[a, EulerPhi[n]/n]], {n, 1, 1000}]; b [From Artur Jasinski (grafix(AT)csl.pl), Nov 05 2008]
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PROGRAM
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(PARI) {a(n)= local(m, c); if(n<=1, 4*(n==1), c=1; m=4; while( c<n, m++; if(!issquarefree(m), c++)); m)} /* Michael Somos Apr 29 2005 */
(MAGMA) [ n : n in [1..1000] | not IsSquarefree(n) ];
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CROSSREFS
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Complement of A005117. Cf. A001694.
Adjacent sequences: A013926 A013927 A013928 this_sequence A013930 A013931 A013932
Sequence in context: A144794 A047462 A020671 this_sequence A162966 A034043 A053443
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KEYWORD
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nonn,easy
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AUTHOR
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Henri Lifchitz (100637.64(AT)CompuServe.COM)
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EXTENSIONS
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More terms from Erich Friedman (erich.friedman(AT)stetson.edu).
More terms from Franz Vrabec (franz.vrabec(AT)planetuniqa.at), Aug 13 2005
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