A008966 a(n) = 1 if n is squarefree, otherwise 0.

1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0

Offset: 1

Author

N. J. A. Sloane

Keywords

• easy
• nonn
• mult

Comments

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3, 1).

The infinite lower triangular matrix with A008966 on the main diagonal and the rest zeros is the square of triangle A143255. - Gary W. Adamson, Aug 02 2008

Links

• Daniel Forgues, Table of n, a(n) for n = 1..100000
• Eric Weisstein's World of Mathematics, Moebius Function
• Index entries for characteristic functions
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A005117, A008836 (Dirichlet inverse), A013928 (partial sums).

Cf. A179211, A179212, A179213, A179214, A179215.

Cf. A020639, A073576, A087188.

Cf. A124010, A212793, A085357, A156552.

Parity of A002033.

Cf. A047999, A036987

Cf. A082020 (Dgf at s=2), A157289 (Dgf at s=3), A157290 (Dgf at s=4).

Programs

Haskell

a008966 = abs . a008683
-- Reinhard Zumkeller, Dec 13 2015, Dec 15 2014, May 27 2012, Jan 25 2012

Magma

[ Abs(MoebiusMu(n)) : n in [1..100]];

Maple

A008966 := proc(n) if numtheory[issqrfree](n) then 1 ; else 0 ; end if; end proc: # R. J. Mathar, Mar 14 2011

Mathematica

A008966[n_] := Abs[MoebiusMu[n]]; Table[A008966[n], {n, 100}] (* Enrique Pérez Herrero, Apr 15 2010 *)
Table[If[SquareFreeQ[n],1,0],{n,100}] (* or *) Boole[SquareFreeQ/@ Range[ 100]] (* Harvey P. Dale, Feb 28 2015 *)

MuPAD

func(abs(numlib::moebius(n)), n):

PARI

a(n)=if(n<1,0,direuler(p=2,n,1+X))[n]

PARI

a(n)=issquarefree(n) \\ Michel Marcus, Feb 22 2015

Python

from sympy import factorint
def A008966(n): return int(max(factorint(n).values(),default=1)==1) # Chai Wah Wu, Apr 05 2023

Formula

Dirichlet g.f.: zeta(s)/zeta(2s).

a(n) = abs(mu(n)), where mu is the Moebius function (A008683).

a(n) = 0^(bigomega(n) - omega(n)), where bigomega(n) and omega(n) are the numbers of prime factors of n with and without repetition (A001222, A001221, A046660). - Reinhard Zumkeller, Apr 05 2003

Multiplicative with p^e -> 0^(e - 1), p prime and e > 0. - Reinhard Zumkeller, Jul 15 2003

a(n) = 0^(A046951(n) - 1). - Reinhard Zumkeller, May 20 2007

a(n) = 1 - A107078(n). - Reinhard Zumkeller, Oct 03 2008

a(n) = floor(rad(n)/n), where rad() is A007947. - Enrique Pérez Herrero, Nov 13 2009

A175046(n) = a(n)*A073311(n). - Reinhard Zumkeller, Apr 05 2010

a(n) = floor(A000005(n^2)/A007425(n)). - Enrique Pérez Herrero, Apr 15 2010

a(A005117(n)) = 1; a(A013929(n)) = 0; a(n) = A013928(n + 1) - A013928(n). - Reinhard Zumkeller, Jul 05 2010

a(n) * A112526(n) = A063524(n). - Reinhard Zumkeller, Sep 16 2011

a(n) = mu(n) * lambda(n) = A008836(n) * A008683(n). - Enrique Pérez Herrero, Nov 29 2013

a(n) = Sum_{d|n} 2^omega(d)*mu(n/d). - Geoffrey Critzer, Feb 22 2015

a(n) = A085357(A156552(n)). - Antti Karttunen, Mar 06 2017

Limit_{n->oo} (1/n)*Sum_{j=1..n} a(j) = 6/Pi^2. - Andres Cicuttin, Aug 13 2017

a(1) = 1; a(n) = -Sum_{d|n, d < n} (-1)^bigomega(n/d) * a(d). - Ilya Gutkovskiy, Mar 10 2021

Extensions

Deleted an unclear comment. - N. J. A. Sloane, May 30 2021