A368929 Dirichlet g.f.: zeta(s-2)^2 * (1 - 2^(3-s)) / zeta(s).

1, -1, 17, -16, 49, -17, 97, -112, 225, -49, 241, -272, 337, -97, 833, -640, 577, -225, 721, -784, 1649, -241, 1057, -1904, 1825, -337, 2673, -1552, 1681, -833, 1921, -3328, 4097, -577, 4753, -3600, 2737, -721, 5729, -5488, 3361, -1649, 3697, -3856, 11025, -1057, 4417

Offset

1,3

Comments

Dirichlet convolution of A007434 and A162395.

Links

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

Formula

Sum_{k=1..n} a(k) ~ log(2) * n^3 / (3*zeta(3)).

Multiplicative with a(2^e) = -(3*e-2)*2^(2*e-2), and a(p^e) = p^(2*e)*(1 + e*(1-1/p^2)) for an odd prime p. - Amiram Eldar, Jan 12 2024

Mathematica

Table[Sum[Sum[d^2 * MoebiusMu[k/d], {d, Divisors[k]}] * (-1)^(n/k + 1) * n^2/k^2, {k, Divisors[n]}], {n, 1, 100}]
f[p_, e_] := p^(2*e)*(1 + e*(1 - 1/p^2)); f[2, e_] := -(3*e - 2)*2^(2*e - 2); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 12 2024 *)

Prog

(PARI) a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; if(p == 2, -(3*e-2)*2^(2*e-2), p^(2*e)*(1 + e*(1-1/p^2)))); } \\ Amiram Eldar, Jan 12 2024

Crossrefs

Cf. A048272, A332794.

Cf. A007434, A162395, A360428.

Sequence in context: A023459 A004458 A082123 * A273973 A172091 A371997

Adjacent sequences: A368926 A368927 A368928 * A368930 A368931 A368932

Keyword

sign,mult

Author

Vaclav Kotesovec, Jan 12 2024

Status

approved