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

1,1

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

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

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 *)

Prog

(MuPAD) func(abs(numlib::moebius(n)), n):
(Magma) [ Abs(MoebiusMu(n)) : n in [1..100]];
(PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1+X))[n]
(PARI) a(n)=issquarefree(n) \\ Michel Marcus, Feb 22 2015
(Haskell)
a008966 = abs . a008683
-- Reinhard Zumkeller, Dec 13 2015, Dec 15 2014, May 27 2012, Jan 25 2012
(Python)
from sympy import factorint
def A008966(n): return int(max(factorint(n).values(), default=1)==1) # Chai Wah Wu, Apr 05 2023

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).

Sequence in context: A293233 A302050 A008683 * A080323 A157657 A359155

Adjacent sequences: A008963 A008964 A008965 * A008967 A008968 A008969

Keyword

easy,nonn,mult

Author

N. J. A. Sloane

Extensions

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

Status

approved