A035171 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = -19.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• mult

Comments

From Jianing Song, Sep 07 2018: (Start)

Half of the number of integer solutions to x^2 + x*y + 5*y^2 = n.

Inverse Moebius transform of A011585. (End)

Coefficients of Dedekind zeta function for the quadratic number field of discriminant -19. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Links

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

Crossrefs

Cf. A028641.

Moebius transform gives A011585.

Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.

Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.

Programs

Mathematica

a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-19, #] &]]; Table[a[n], {n, 1, 100}] (* G. C. Greubel, Jul 17 2018 *)

PARI

m = -19; direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

Formula

From Jianing Song, Sep 07 2018: (Start)

a(n) is multiplicative with a(19^e) = 1, a(p^e) = (1 + (-1)^e) / 2 if Kronecker(-19, p) = -1, a(p^e) = e + 1 if Kronecker(-19, p) = 1.

G.f.: Sum_{k>0} Kronecker(-19, k) * x^k / (1 - x^k).

A028641(n) = 2 * a(n) unless n = 0.

(End)

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(19) = 0.720730... . - Amiram Eldar, Oct 11 2022