A328254 - OEIS

%I #22 Dec 02 2020 03:16:00

%S 1,-7,-13,14,-31,91,-57,-8,39,217,-133,-182,-183,399,403,0,-307,-273,

%T -381,-434,741,931,-553,104,155,1281,-27,-798,-871,-2821,-993,0,1729,

%U 2149,1767,546,-1407,2667,2379,248,-1723,-5187,-1893,-1862,-1209,3871,-2257,0,399,-1085

%N Dirichlet g.f.: 1 / (zeta(s) * zeta(s-1) * zeta(s-2)).

%C Dirichlet inverse of A001001.

%H Amiram Eldar, <a href="/A328254/b328254.txt">Table of n, a(n) for n = 1..10000</a>

%F a(1) = 1; a(n) = -Sum_{d|n, d<n} A001001(n/d) * a(d).

%F Multiplicative with a(p^e) = -(p^2+p+1) if e=1, p^3+p^2+p if e=2, -p^3 if e=3, and 0 otherwise. - _Amiram Eldar_, Dec 02 2020

%t a[1] = 1; a[n_] := -Sum[Sum[j DivisorSigma[1, j], {j, Divisors[n/d]}] a[d], {d, Most @Divisors[n]}]; Table[a[n], {n, 1, 50}]

%t f[p_, e_] := Which[e==1, -(p^2+p+1), e==2, p^3+p^2+p, e==3, -p^3, e>3, 0]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Dec 02 2020 *)

%o (PARI) seq(n)={dirdiv(vector(n,n,n==1), vector(n,n,sumdivmult(n, d, sigma(d)*d)))} \\ _Andrew Howroyd_, Oct 25 2019

%Y Cf. A001001, A046101 (positions of 0's), A046692.

%K sign,mult

%O 1,2

%A _Ilya Gutkovskiy_, Oct 22 2019