A035186 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 3.

1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 2, 1, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 1, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1

Offset

1,11

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Formula

From Amiram Eldar, Oct 17 2022: (Start)

a(n) = Sum_{d|n} Kronecker(3, d).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(2+sqrt(3)) / (3*sqrt(3)) = 0.506897... . (End)

Multiplicative with a(3^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(3, p) = -1 (p is in A038875), and a(p^e) = e+1 if Kronecker(3, p) = 1 (p is in A097933). - Amiram Eldar, Nov 20 2023

Mathematica

a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[3, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)

Prog

(PARI) my(m=3); direuler(p=2, 101, 1/(1-(kronecker(m, p)*(X-X^2))-X))
(PARI) a(n) = sumdiv(n, d, kronecker(3, d)); \\ Amiram Eldar, Nov 20 2023

Crossrefs

Cf. A038875, A097933.

Sequence in context: A035179 A035161 A352565 * A035194 A245818 A161491

Adjacent sequences: A035183 A035184 A035185 * A035187 A035188 A035189

Keyword

nonn,easy,mult

Author

N. J. A. Sloane

Status

approved