A307425 Dirichlet g.f.: zeta(s) / (zeta(2*s) * zeta(3*s)).

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

Offset

1

Comments

Dirichlet convolution of A212793 and A271102.

Links

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

Vaclav Kotesovec, Graph - the asymptotic ratio.

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

Wikipedia, Dirichlet series.

Formula

Sum_{k=1..n} a(k) ~ 6*n / (Pi^2 * zeta(3)).

Multiplicative with a(p) = 1, a(p^e) = -1 if e = 3 or 4, and 0 if e = 2 or e >= 5. - Amiram Eldar, Dec 25 2022

Mathematica

nmax = 100; A271102 = Table[DivisorSum[n, Abs[MoebiusMu[#]]*MoebiusMu[n/#] &], {n, 1, nmax}]; Table[DivisorSum[n, Boole[Max[FactorInteger[#][[All, 2]]] < 3] * A271102[[n/#]] &], {n, 1, nmax}]
f[p_, e_] := Switch[e, 1, 1, 3, -1, 4, -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)*(1-X^3))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

Crossrefs

Cf. A056624, A210826, A212793, A271102, A299406, A307420 (Dirichlet inverse).

Sequence in context: A181183 A295316 A014677 * A210826 A307421 A299406

Adjacent sequences: A307422 A307423 A307424 * A307426 A307427 A307428

Keyword

sign,mult

Author

Vaclav Kotesovec, Apr 08 2019

Status

approved