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A013929
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Numbers that are not squarefree. Numbers that are divisible by a square greater than 1. The complement of A005117.
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155
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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
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OFFSET
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1,1
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COMMENTS
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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, Dec 16 2005
Numbers n such that sum(d|n, d/phi(d)*mu(n/d))=0 - Benoit Cloitre, Apr 28 2002
Also n such there is at least one x<n such that A007913(x)=A007913(n) - Benoit Cloitre, 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, 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, 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, Feb 06 2009]
Number of prime divisors of (n+1)<number of nonprime divisors of (n+1). [From Juri-Stepan Gerasimov, Nov 10 2009]
Orders for which at least one non-cyclic finite abelian group exists: A000688(a(n))>1. This follows from the fact that not all exponents in the prime number factorization of a(n) are 1 (moebius(a(n))=0). The number of such groups of order a(n) is A192005(n) = A000688(a(n))-1. [From Wolfdieter Lang Jul 29 2011]
Subsequence of A193166; A192280(a(n)) = 0. [Reinhard Zumkeller, Aug 26 2011]
It appears that a(n) = n such that product(ithprime(k) ,k=1..n) mod n <>0. See Maple code. [From Gary Detlefs, Dec 07 2011]
A008477(a(n)) > 1. [Reinhard Zumkeller, Feb 17 2012]
A057918(a(n)) > 0. [Reinhard Zumkeller, Mar 27 2012]
Numbers for which gcd(n,n')>1, where n' is the arithmetic derivative of n. [Paolo P. Lava, Apr 24 2012]
Sum(n>=1, 1/a(n)^s) = (Zeta(s)* (Zeta(2*s)-1))/Zeta(2*s). - Enrique Pérez Herrero, Jul 07 2012
A056170(a(n)) > 0. - Reinhard Zumkeller, Dec 29 2012
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REFERENCES
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L. Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829, 2012. - From N. J. A. Sloane, Jan 01 2013
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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, Smarandache Near-to-Primorial Function, Squarefree, Squareful, Moebius Function.
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FORMULA
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A008966(a(n)) = 0. [Reinhard Zumkeller, Apr 22 2012]
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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, May 04 2009]
t:= n-> product(ithprime(k), k=1..n): for n from 1 to 160 do (if t(n) mod n <>0) then print(n) fi od; [Gary Detlefs, Dec 07 2011]
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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 &] (* Robert G. Wilson v, 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, Nov 05 2008]
Cases[Range[160], n_ /; !SquareFreeQ[n]] (* Jean-François Alcover, Mar 21 2011 *)
Select[Range@160, ! SquareFreeQ[#] &] (* Robert G. Wilson v, Jul 21 2012 *)
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PROG
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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) ];
(Haskell)
a013929 n = a013929_list !! (n-1)
a013929_list = filter ((== 0) . a008966) [1..]
-- Reinhard Zumkeller, Apr 22 2012
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CROSSREFS
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Complement of A005117. Cf. A001694, A038109.
Cf. A130897 (subsequence).
Cf. A190641 (subsequence).
Sequence in context: A020671 A171598 * A162966 A190641 A034043 A053443
Adjacent sequences: A013926 A013927 A013928 * A013930 A013931 A013932
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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.
More terms from Franz Vrabec, Aug 13 2005
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STATUS
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approved
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