A369100 Dirichlet g.f.: zeta(s)^3 * (1 - 2^(1-s))^2.

1, -1, 3, -2, 3, -3, 3, -2, 6, -3, 3, -6, 3, -3, 9, -1, 3, -6, 3, -6, 9, -3, 3, -6, 6, -3, 10, -6, 3, -9, 3, 1, 9, -3, 9, -12, 3, -3, 9, -6, 3, -9, 3, -6, 18, -3, 3, -3, 6, -6, 9, -6, 3, -10, 9, -6, 9, -3, 3, -18, 3, -3, 18, 4, 9, -9, 3, -6, 9, -9, 3, -12, 3, -3, 18

Offset: 1

Author

Vaclav Kotesovec, Jan 13 2024

Keywords

• sign
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A007425, A048272, A288571.

Programs

Mathematica

Table[Sum[Sum[-(-1)^d, {d, Divisors[k]}]*(-1)^(n/k+1), {k, Divisors[n]}], {n, 1, 100}]
f[p_, e_] := (e + 1)*(e + 2)/2; f[2, e_] := (e^2 - 5*e + 2)/2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 13 2024 *)

PARI

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e=f[i,2]; if(p == 2, (e^2-5*e+2)/2, (e+1)*(e+2)/2));} \\ Amiram Eldar, Jan 13 2024

Formula

Sum_{k=1..n} a(k) ~ n * log(2)^2.

Multiplicative with a(2^e) = (e^2-5*e+2)/2, and a(p^e) = (e+1)*(e+2)/2 for an odd prime p. - Amiram Eldar, Jan 13 2024