A307420 Dirichlet g.f.: zeta(2*s) * zeta(3*s) / zeta(s).

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

Offset

1

Comments

Dirichlet convolution of A008836 and A010057.

Dirichlet convolution of A210826 and A010052.

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^e) = 1 if e == 0 or 2 (mod 6), -1 if e == 1 (mod 6), and 0 otherwise. - Amiram Eldar, Dec 25 2022

Mathematica

Table[DivisorSum[n, Boole[IntegerQ[#^(1/3)]] * LiouvilleLambda[n/#]&], {n, 1, 100}]
Table[DivisorSum[n, Mod[DivisorSigma[0, #], 3, -1] * Boole[IntegerQ[Sqrt[n/#]]] &], {n, 1, 100}]
f[p_, e_] := Switch[Mod[e, 6], 0, 1, 1, -1, 2, 1, 3, 0, 4, 0, 5, 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/((1+X)*(1-X^3)))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

Crossrefs

Cf. A010052, A010057, A008836, A210826, A307425 (Dirichlet inverse).

Sequence in context: A053864 A189021 A212793 * A129667 A071374 A071025

Adjacent sequences: A307417 A307418 A307419 * A307421 A307422 A307423

Keyword

sign,mult

Author

Vaclav Kotesovec, Apr 08 2019

Status

approved