A328254 Dirichlet g.f.: 1 / (zeta(s) * zeta(s-1) * zeta(s-2)).

1, -7, -13, 14, -31, 91, -57, -8, 39, 217, -133, -182, -183, 399, 403, 0, -307, -273, -381, -434, 741, 931, -553, 104, 155, 1281, -27, -798, -871, -2821, -993, 0, 1729, 2149, 1767, 546, -1407, 2667, 2379, 248, -1723, -5187, -1893, -1862, -1209, 3871, -2257, 0, 399, -1085

Offset: 1

Author

Ilya Gutkovskiy, Oct 22 2019

Keywords

• sign
• mult

Comments

Dirichlet inverse of A001001.

Links

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

Crossrefs

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

Programs

Mathematica

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}]
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 *)

PARI

seq(n)={dirdiv(vector(n,n,n==1), vector(n,n,sumdivmult(n, d, sigma(d)*d)))} \\ Andrew Howroyd, Oct 25 2019

Formula

a(1) = 1; a(n) = -Sum_{d|n, dA001001(n/d) * a(d).

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