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

1,5

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

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

Mathematica

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

Prog

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

Crossrefs

Cf. A028641.

Moebius transform gives A011585.

Cf. A106863 (primes not inert in Q(sqrt(-19))), A191019 (primes decomposing), A191063 (primes remaining inert).

Dedekind zeta functions for imaginary quadratic number fields of discriminants D = -3..-47, -67, -163: A002324, A002654, A035182, A002325, A035179, A035175, this sequence, A035170, A035167, A192013, A035159, A035155, A035151, A035180, A035147, A035143, A318982, A318983.

Dedekind zeta functions for real quadratic number fields of discriminants D = 5..41: A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, A035223.

Sequence in context: A090193 A039974 A039973 * A352567 A088700 A287121

Adjacent sequences: A035168 A035169 A035170 * A035172 A035173 A035174

Keyword

nonn,easy,mult

Author

N. J. A. Sloane

Status

approved