A035183 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -5.

1, 0, 2, 1, 1, 0, 2, 0, 3, 0, 0, 2, 0, 0, 2, 1, 0, 0, 0, 1, 4, 0, 2, 0, 1, 0, 4, 2, 2, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 0, 2, 0, 2, 0, 3, 0, 2, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 6, 1, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 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. A139513, A296923.

Programs

Mathematica

a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-5, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)

PARI

my(m=-5); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

PARI

a(n) = sumdiv(n, d, kronecker(-5, d)); \\ Michel Marcus, Oct 07 2023

Formula

From Amiram Eldar, Oct 17 2022: (Start)

a(n) = Sum_{d|n} Kronecker(-5, d).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(5)) = 0.936641... . (End)

Multiplicative with a(5^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(-5, p) = -1 (p is in A296923), and a(p^e) = e+1 if Kronecker(-5, p) = 1 (p is in A139513). - Amiram Eldar, Nov 20 2023