A369101 Dirichlet g.f.: zeta(s-3)^2 * (1 - 2^(4-s)) / zeta(s).

1, -1, 53, -64, 249, -53, 685, -960, 2133, -249, 2661, -3392, 4393, -685, 13197, -11264, 9825, -2133, 13717, -15936, 36305, -2661, 24333, -50880, 46625, -4393, 76545, -43840, 48777, -13197, 59581, -118784, 141033, -9825, 170565, -136512, 101305, -13717, 232829

Offset: 1

Author

Vaclav Kotesovec, Jan 13 2024

Keywords

• sign
• mult

Comments

In general, for k > 0, if Dirichlet g.f. is zeta(s-k)^2 * (1 - 2^(k+1-s)) / zeta(s), then a(n) ~ log(2) * n^(k+1) / ((k+1) * zeta(k+1)).

Crossrefs

Cf. A048272 (k=0), A332794 (k=1), A368929 (k=2).

Cf. A000578, A059376.

Programs

Mathematica

Table[Sum[DivisorSum[k, #^3*MoebiusMu[k/#]&]*(-1)^(n/k+1)*(n/k)^3, {k, Divisors[n]}], {n, 1, 50}]
f[p_, e_] := p^(3*e-3) * (1 + (e+1)*(p^3-1)); f[2, e_] := -(7*e-6)*8^(e-1); 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, -(7*e-6)*8^(e-1), p^(3*e-3) * (1 + (e+1)*(p^3-1))));} \\ Amiram Eldar, Jan 13 2024

Formula

Sum_{k=1..n} a(k) ~ 45 * log(2) * n^4 / (2*Pi^4).

Multiplicative with a(2^e) = -(7*e-6)*8^(e-1), and a(p^e) = p^(3*e-3) * (1 + (e+1)*(p^3-1)) for an odd prime p. - Amiram Eldar, Jan 13 2024