A369101 - OEIS

%I #13 Jan 13 2024 13:06:50

%S 1,-1,53,-64,249,-53,685,-960,2133,-249,2661,-3392,4393,-685,13197,

%T -11264,9825,-2133,13717,-15936,36305,-2661,24333,-50880,46625,-4393,

%U 76545,-43840,48777,-13197,59581,-118784,141033,-9825,170565,-136512,101305,-13717,232829

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

%C 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)).

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

%F 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

%t Table[Sum[DivisorSum[k, #^3*MoebiusMu[k/#]&]*(-1)^(n/k+1)*(n/k)^3, {k, Divisors[n]}], {n, 1, 50}]

%t 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 *)

%o (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

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

%Y Cf. A000578, A059376.

%K sign,mult

%O 1,3

%A _Vaclav Kotesovec_, Jan 13 2024