A307430 Dirichlet g.f.: zeta(s) / zeta(4*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, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1

Offset

1

Comments

Dirichlet convolution of A227291 and A008966.

The characteristic function of the biquadratefree numbers (A046100). - Amiram Eldar, Dec 27 2022

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) ~ 90*n/Pi^4.

Multiplicative with a(p^e) = 1 if e <= 3, and 0 otherwise. - Amiram Eldar, Dec 27 2022

Mathematica

nmax = 100; A227291 = Abs[Table[DivisorSum[n, Abs[MoebiusMu[#]]*MoebiusMu[n/#] &], {n, 1, nmax}]]; Table[DivisorSum[n, Abs[MoebiusMu[n/#]] * A227291[[#]] &], {n, 1, nmax}]
a[n_] := If[Max[FactorInteger[n][[;; , 2]]] < 4, 1, 0]; Array[a, 100] (* Amiram Eldar, Dec 27 2022 *)

Prog

(PARI) for(n=1, 100, print1(direuler(p=2, n, (1-X^4)/(1-X))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

Crossrefs

Cf. A008966, A046100, A212793, A219009 (Dirichlet inverse), A227291.

Sequence in context: A340369 A307445 A363552 * A053865 A189022 A370598

Adjacent sequences: A307427 A307428 A307429 * A307431 A307432 A307433

Keyword

nonn,mult

Author

Vaclav Kotesovec, Apr 08 2019

Status

approved