A035177 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -13.

1, 0, 0, 1, 0, 0, 2, 0, 1, 0, 2, 0, 1, 0, 0, 1, 2, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 3, 0, 0, 1, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 1, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 4, 0, 0, 0, 1

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A296926.

Programs

Mathematica

a[n_] := DivisorSum[n, KroneckerSymbol[-13, #] &]; Array[a, 100] (* Amiram Eldar, Nov 17 2023 *)

PARI

my(m = -13); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

PARI

a(n) = sumdiv(n, d, kronecker(-13, d)); \\ Amiram Eldar, Nov 17 2023

Formula

From Amiram Eldar, Nov 17 2023 (Start)

a(n) = Sum_{d|n} Kronecker(-13, d).

Multiplicative with a(13^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(-13, p) = -1, and a(p^e) = e+1 if Kronecker(-13, p) = 1 (p is in A296926).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(13)) = 0.5808806... . (End)