A307423 Dirichlet g.f.: zeta(2*s) / zeta(3*s).

1, 0, 0, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1

Offset

1

Comments

Dirichlet convolution of A212793 and A008836.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Dirichlet Generating Function.

Wikipedia, Dirichlet series.

Formula

Multiplicative with a(p) = 0, and a(p^e) = (-1)^e for e >= 2. - Amiram Eldar, Dec 25 2022

Mathematica

Table[DivisorSum[n, Boole[Max[FactorInteger[#][[All, 2]]] < 3] * LiouvilleLambda[n/#]&], {n, 1, 100}]
f[p_, e_] := (-1)^e; f[p_, 1] := 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 25 2022 *)

Prog

(PARI) for(n=1, 100, print1(direuler(p=2, n, (1-X^3)/(1-X^2))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020
(PARI)
A212793(n) = factorback(apply(e->(e<=2), factor(n)[, 2]));
A307423(n) = sumdiv(n, d, ((-1)^bigomega(d))*A212793(n/d)); \\ Antti Karttunen, Jul 14 2022

Crossrefs

Cf. A008836, A010052, A056624, A212793, A307424.

Cf. A112526 (absolute values).

Sequence in context: A354868 A075802 A112526 * A355684 A355683 A373374

Adjacent sequences: A307420 A307421 A307422 * A307424 A307425 A307426

Keyword

sign,mult

Author

Vaclav Kotesovec, Apr 08 2019

Extensions

Data section extended up to a(108) by Antti Karttunen, Jul 14 2022

Status

approved