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A178448 Dirichlet inverse of A001160, sigma_5.

%I A178448 #25 Sep 16 2020 05:05:42
%S A178448 1,-33,-244,32,-3126,8052,-16808,0,243,103158,-161052,-7808,-371294,
%T A178448 554664,762744,0,-1419858,-8019,-2476100,-100032,4101152,5314716,
%U A178448 -6436344,0,3125,12252702,0,-537856,-20511150,-25170552,-28629152,0,39296688,46855314,52541808
%N A178448 Dirichlet inverse of A001160, sigma_5.
%H A178448 Amiram Eldar, <a href="/A178448/b178448.txt">Table of n, a(n) for n = 1..10000</a>
%F A178448 Dirichlet g.f.: 1/(zeta(s)*zeta(s-5)). - _R. J. Mathar_, Mar 10 2011
%F A178448 a(n) = Sum_{d|n} mu(n/d)*mu(d)*d^5. - _Ilya Gutkovskiy_, Nov 06 2018
%F A178448 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
%t A178448 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 *)
%t A178448 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 *)
%o A178448 (PARI) A178448_vec(len)={
%o A178448         a063524=vector(len) ; a063524[1] = 1 ;
%o A178448         a001160=direuler(p=2,len, 1/(1-p^5*X)/(1-X)) ;
%o A178448         dirdiv(a063524,a001160) ;}
%o A178448 { A178448_vec(70) } /* _R. J. Mathar_, Mar 10 2011 */
%o A178448 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*moebius(d)*d^5); \\ _Michel Marcus_, Nov 06 2018
%o A178448 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - X)*(1 - p^5*X))[n], ", ")) \\ _Vaclav Kotesovec_, Sep 16 2020
%Y A178448 Cf. A001160, A053825, A053826.
%K A178448 sign,mult
%O A178448 1,2
%A A178448 _R. J. Mathar_, Dec 22 2010
%E A178448 More terms from _Amiram Eldar_, Sep 16 2020