A307445 Dirichlet g.f.: 1 / (zeta(s) * zeta(2*s)).

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

Offset

1

Comments

Dirichlet convolution of A008966 and A007427.

Dirichlet convolution of A008683 and A271102.

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) = a(p^2) = -1, a(p^3) = 1, and a(p^e) = 0 for e >= 4. - Amiram Eldar, Dec 25 2022

Mathematica

nmax = 100; A271102 = Table[DivisorSum[n, Abs[MoebiusMu[#]]*MoebiusMu[n/#] &], {n, 1, nmax}]; Table[DivisorSum[n, MoebiusMu[n/#]*A271102[[#]] &], {n, 1, nmax}]
nmax = 100; A007427 = Table[DivisorSum[n, MoebiusMu[#]*MoebiusMu[n/#] &], {n, 1, nmax}]; Table[DivisorSum[n, Abs[MoebiusMu[n/#]]*A007427[[#]] &], {n, 1, nmax}]
f[p_, e_] := 0; f[p_, 1] = f[p_, 2] = -1; f[p_, 3] = 1; 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^2)*(1-X))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

Crossrefs

Cf. A007427, A008683, A008966, A271102.

Sequence in context: A167850 A167851 A340369 * A363552 A307430 A053865

Adjacent sequences: A307442 A307443 A307444 * A307446 A307447 A307448

Keyword

sign,mult

Author

Vaclav Kotesovec, Apr 08 2019

Status

approved