A307428 Dirichlet g.f.: zeta(2*s) / (zeta(s) * zeta(3*s)).

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

Offset: 1

Author

Vaclav Kotesovec, Apr 08 2019

Keywords

• sign
• mult

Comments

Dirichlet convolution of A307423 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, A010052, A056624, A307423.

Programs

Mathematica

nmax = 100; A307423 = Table[DivisorSum[n, Boole[Max[FactorInteger[#][[All, 2]]] < 3] * LiouvilleLambda[n/#]&], {n, 1, nmax}]; Table[DivisorSum[n, MoebiusMu[#] * A307423[[n/#]] &], {n, 1, nmax}]
f[p_, e_] := 2*(-1)^e; f[p_, 1] := -1; f[p_, 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^3)/(1+X))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

Formula

Multiplicative with a(p) = -1, a(p^2) = 1, and a(p^e) = 2*(-1)^e for e >= 3. - Amiram Eldar, Dec 25 2022