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A013929 Numbers that are not squarefree. Numbers that are divisible by a square greater than 1. The complement of A005117. 214
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; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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, 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) = k such that phi(k)/k = phi(m)/m for some m < k. - 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. - Mats Granvik, Feb 06 2009

Number of prime divisors of (n+1) is less than the number of nonprime divisors of (n+1). - 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. - 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_{k=1..n} (prime(k)) mod n <> 0. See Maple code. - 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

a(n) = n such that A001221(n) != A001222(n). - Felix Fröhlich, Aug 13 2014

Numbers whose sum of divisors is greater than the sum of unitary divisors: A000203(a(n)) > A034448(a(n)). - Paolo P. Lava, Oct 08 2014

Numbers n that A001222(n) > A001221(n), since in this case at least one prime factor of n occurs more than once, what implies that n is divisible by at least one perfect square > 1. - Carlos Eduardo Olivieri, Aug 02 2015

Lexicographically least sequence such that each entry has a positive even number of proper divisors _not_ occurring in the sequence, cf. the sieve characterization of A005117. - Glen Whitney, Aug 30 2015

There are arbitrarily long runs of consecutive terms. Record runs start at 4, 8, 48, 242, ... (A045882). - Ivan Neretin, Nov 07 2015

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

Louis Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829 [math.NT], 2012.

Srinivasa 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.

FORMULA

A008966(a(n)) = 0. - Reinhard Zumkeller, Apr 22 2012

a(n) ~ n/k, where k = 1 - 1/zeta(2) = 1 - 6/Pi^2 = A229099. - Charles R Greathouse IV, Sep 13 2013

A001222(a(n)) > A001221(a(n)). - Carlos Eduardo Olivieri, Aug 02 2015

MAPLE

a := n -> `if`(numtheory[mobius](n)=0, n, NULL); seq(a(i), i=1..160); # 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

MATHEMATICA

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 *)

Cases[Range[160], n_ /; !SquareFreeQ[n]] (* Jean-François Alcover, Mar 21 2011 *)

Select[Range@160, ! SquareFreeQ[#] &] (* Robert G. Wilson v, Jul 21 2012 *)

Select[Range@160, PrimeOmega[#] > PrimeNu[#] &] (* Carlos Eduardo Olivieri, Aug 02 2015 *)

Select[Range[200], MoebiusMu[#] == 0 &] (* Alonso del Arte, Nov 07 2015 *)

PROG

(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

(PARI) for(n=1, 1e3, if(omega(n)!=bigomega(n), print1(n, ", "))) \\ Felix Fröhlich, Aug 13 2014

CROSSREFS

Complement of A005117. Cf. A001694, A038109.

Cf. A130897 (subsequence).

Cf. A190641 (subsequence).

Partitions into: A114374, A256012.

Sequence in context: A171598 A259241 * A162966 A190641 A034043 A278517

Adjacent sequences:  A013926 A013927 A013928 * A013930 A013931 A013932

KEYWORD

nonn,easy

AUTHOR

Henri Lifchitz

EXTENSIONS

More terms from Erich Friedman

More terms from Franz Vrabec, Aug 13 2005

STATUS

approved

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Last modified December 3 18:47 EST 2016. Contains 278745 sequences.