A307427 Dirichlet g.f.: zeta(3*s) / (zeta(s) * zeta(2*s)).

1, -1, -1, -1, -1, 1, -1, 2, -1, 1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, 1, 1, -1, -2, -1, 1, 2, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, 1, 1, -2, -1, -1, -1, 1, 1, 1, -1, 1, -1, 1, 1, 1, -1, -2, 1, -2, 1, 1, -1, -1, -1, 1, 1, 2, 1, -1, -1, 1, 1, -1, -1, -2, -1, 1

Offset: 1

Author

Vaclav Kotesovec, Apr 08 2019

Keywords

• sign
• mult

Comments

Dirichlet convolution of A210826 and A271102.

Dirichlet convolution of A307424 and A008683.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Dirichlet Generating Function.
• Wikipedia, Dirichlet series.

Crossrefs

Cf. A008683, A010057, A210826, A271102, A307424.

Programs

Mathematica

nmax = 100; A271102 = Table[DivisorSum[n, Abs[MoebiusMu[#]] * MoebiusMu[n/#] &], {n, 1, nmax}]; Table[DivisorSum[n, Mod[DivisorSigma[0, n/#], 3, -1] * A271102[[#]] &], {n, 1, nmax}]
nmax = 100; A307424 = Table[DivisorSum[n, Abs[MoebiusMu[#]] * Mod[DivisorSigma[0, n/#], 3, -1]&], {n, 1, nmax}]; Table[DivisorSum[n, MoebiusMu[#] * A307424[[n/#]] &], {n, 1, nmax}]
f[p_, e_] := If[Divisible[e, 3], 2, -1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 25 2022 *)

PARI

for(n=1, 100, print1(direuler(p=2, n, (1-X)*(1-X^2)/(1-X^3))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

Formula

Multiplicative with a(p^e) = 2 if e == 0 (mod 3), and -1 otherwise. - Amiram Eldar, Dec 25 2022