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A063659 Numbers 1 <= m <= n for which GCD(m,n) is not divisible by a square greater than 1. 10
1, 2, 3, 3, 5, 6, 7, 6, 8, 10, 11, 9, 13, 14, 15, 12, 17, 16, 19, 15, 21, 22, 23, 18, 24, 26, 24, 21, 29, 30, 31, 24, 33, 34, 35, 24, 37, 38, 39, 30, 41, 42, 43, 33, 40, 46, 47, 36, 48, 48, 51, 39, 53, 48, 55, 42, 57, 58, 59, 45, 61, 62, 56, 48, 65, 66, 67, 51, 69, 70, 71, 48 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Equals Möbius transform of A001615. - Gary W. Adamson, May 23 2008

The absolute values of the Dirichlet inverse of A007913. - R. J. Mathar, Dec 22 2010

LINKS

Harry J. Smith, Table of n, a(n) for n=1..2000

Wolfgang Schramm, The Fourier transform of functions of the greatest common divisor, Electronic Journal of Combinatorial Number Theory A50 (8(1)), 2008.

FORMULA

a(n) = n - A063658(n).

Multiplicative with a(p) = p and a(p^e) = p^e-p^(e-2), e>1. - Vladeta Jovovic, Jul 26 2001

a(n) = sum(d|n, phi(d)*mu(n/d)^2 ), Dirichlet convolution of A000010 and A008966. - Benoit Cloitre, Sep 08 2002

a(n) = sum(k=1,n,mu(gcd(n,k))^2). - Benoit Cloitre, Jun 14 2007

Dirichlet g.f. zeta(s-1)/zeta(2s). - R. J. Mathar, Feb 27 2011

a(n) = Sum_{k=1..n} psi(GCD(k,n)) * Cos(2*Pi*k/n), where psi is A001615. - Enrique Pérez Herrero, Jan 18 2013

EXAMPLE

For n=12 we find only GCD(4,12), GCD(8,12) and GCD(12,12) divisible by 4, so a(12)=9.

MATHEMATICA

nn = 72; f[list_, i_] := list[[i]]; a =Table[If[Max[FactorInteger[n][[All, 2]]] < 2, 1, 0], {n, 1, nn}]; b =Table[EulerPhi[n], {n, 1, nn}]; Table[

DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Feb 22 2015 *)

PROG

(PARI) a(n)=sum(k=1, n, moebius(gcd(n, k))^2) \\ Benoit Cloitre, Jun 14 2007

(PARI) { for (n=1, 2000, a=1; for (m=2, n, if (issquarefree(gcd(m, n)), a++)); write("b063659.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 27 2009

CROSSREFS

Cf. A001615.

Sequence in context: A097247 A097246 A277886 * A255563 A115350 A262882

Adjacent sequences:  A063656 A063657 A063658 * A063660 A063661 A063662

KEYWORD

mult,nonn

AUTHOR

Floor van Lamoen, Jul 24 2001

EXTENSIONS

More terms from Vladeta Jovovic and Dean Hickerson, Jul 26 2001

STATUS

approved

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Last modified August 19 01:37 EDT 2017. Contains 290788 sequences.