Search: id:A013929
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%I A013929
%S A013929 4,8,9,12,16,18,20,24,25,27,28,32,36,40,44,45,48,49,50,52,54,56,60,63,
%T A013929 64,68,72,75,76,80,81,84,88,90,92,96,98,99,100,104,108,112,116,117,120,
%U A013929 121,124,125,126,128,132,135,136,140,144,147,148,150,152,153,156,160
%N A013929 Numbers that are not square-free. Numbers that are divisible by a square 
               greater than 1. The complement of A005117.
%C A013929 Sometimes misnamed squareful numbers, but officially those are given 
               by A001694.
%C A013929 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
%C A013929 Numbers n such that sum(d|n, d/phi(d)*mu(n/d))=0 - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 28 2002
%C A013929 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
%C A013929 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
%C A013929 Numbers n such there is a solution 0<x<n to x^2==0 mod(n). - Franz Vrabec 
               (franz.vrabec(AT)aon.at), Aug 13 2005
%C A013929 Numbers n such that moebius(n) = 0.
%C A013929 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]
%C A013929 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]
%C A013929 Number of prime divisors of (n+1)<number of nonprime divisors of (n+1). 
               [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 10 2009]
%H A013929 T. D. Noe, <a href="b013929.txt">Table of n, a(n) for n=1..1000</a>
%H A013929 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/
               Cpaper4/page1.htm">Irregular numbers</a>, J. Indian Math. Soc. 5 
               (1913) 105-106.
%H A013929 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SmarandacheNear-to-PrimorialFunction.html">Link to a section of The 
               World of Mathematics.</a>
%H A013929 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Squarefree.html">Link to a section of The World of Mathematics.</
               a>
%H A013929 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Squareful.html">Link to a section of The World of Mathematics.</a>
%H A013929 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               MoebiusFunction.html">Moebius Function</a>
%p A013929 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]
%t A013929 Union[ Flatten[ Table[ n i^2, {i, 2, 20}, {n, 1, 400/i^2} ] ] ]
%t A013929 Select[ Range[2, 160], (Union[Last /@ FactorInteger[ # ]][[ -1]] > 1) 
               == True &] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 11 2005)
%t A013929 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]
%o A013929 (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 */
%o A013929 (MAGMA) [ n : n in [1..1000] | not IsSquarefree(n) ];
%Y A013929 Complement of A005117. Cf. A001694.
%Y A013929 Sequence in context: A144794 A047462 A020671 this_sequence A162966 A034043 
               A053443
%Y A013929 Adjacent sequences: A013926 A013927 A013928 this_sequence A013930 A013931 
               A013932
%K A013929 nonn,easy,new
%O A013929 1,1
%A A013929 Henri Lifchitz (100637.64(AT)CompuServe.COM)
%E A013929 More terms from Erich Friedman (erich.friedman(AT)stetson.edu).
%E A013929 More terms from Franz Vrabec (franz.vrabec(AT)aon.at), Aug 13 2005

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