A178448 Dirichlet inverse of A001160, sigma_5.

1, -33, -244, 32, -3126, 8052, -16808, 0, 243, 103158, -161052, -7808, -371294, 554664, 762744, 0, -1419858, -8019, -2476100, -100032, 4101152, 5314716, -6436344, 0, 3125, 12252702, 0, -537856, -20511150, -25170552, -28629152, 0, 39296688, 46855314, 52541808

Offset: 1

Author

R. J. Mathar, Dec 22 2010

Keywords

• sign
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001160, A053825, A053826.

Programs

Mathematica

a[1] = 1; a[n_] := a[n] = -Sum[ DivisorSigma[5, n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 29}] (* Jean-François Alcover, Jun 24 2013 *)
f[p_, e_] := If[e == 1, -p^5 - 1, If[e == 2, p^5, 0]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

PARI

A178448_vec(len)={
        a063524=vector(len) ; a063524[1] = 1 ;
        a001160=direuler(p=2,len, 1/(1-p^5*X)/(1-X)) ;
        dirdiv(a063524,a001160) ;}
{ A178448_vec(70) } /* R. J. Mathar, Mar 10 2011 */

PARI

a(n) = sumdiv(n, d, moebius(n/d)*moebius(d)*d^5); \\ Michel Marcus, Nov 06 2018

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 - X)*(1 - p^5*X))[n], ", ")) \\ Vaclav Kotesovec, Sep 16 2020

Formula

Dirichlet g.f.: 1/(zeta(s)*zeta(s-5)). - R. J. Mathar, Mar 10 2011

a(n) = Sum_{d|n} mu(n/d)*mu(d)*d^5. - Ilya Gutkovskiy, Nov 06 2018

Multiplicative with a(p) = -1 - p^5, a(p^2) = p^5, and a(p^e) = 0 for e>=3. - Amiram Eldar, Sep 16 2020

Extensions

More terms from Amiram Eldar, Sep 16 2020