A035189 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 7.

1, 2, 2, 3, 0, 4, 1, 4, 3, 0, 0, 6, 0, 2, 0, 5, 0, 6, 2, 0, 2, 0, 0, 8, 1, 0, 4, 3, 2, 0, 2, 6, 0, 0, 0, 9, 2, 4, 0, 0, 0, 4, 0, 0, 0, 0, 2, 10, 1, 2, 0, 0, 2, 8, 0, 4, 4, 4, 2, 0, 0, 4, 3, 7, 0, 0, 0, 0, 0, 0, 0, 12, 0, 4, 2, 6, 0, 0, 0, 0, 5

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A003632, A038878.

Programs

Mathematica

a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[7, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)

PARI

my(m=7); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

PARI

a(n) = sumdiv(n, d, kronecker(7, d)); \\ Amiram Eldar, Nov 20 2023

Formula

From Amiram Eldar, Oct 17 2022: (Start)

a(n) = Sum_{d|n} Kronecker(7, d).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(8+3*sqrt(7)) / sqrt(7) = 2.0929097... . (End)

Multiplicative with a(7^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(7, p) = -1 (p is in A003632), and a(p^e) = e+1 if Kronecker(7, p) = 1 (p is in A038878 \ {7}). - Amiram Eldar, Nov 20 2023