A035171 - OEIS

%I #24 Oct 11 2022 06:16:06

%S 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,

%T 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,

%U 0,0,0,0,2,0,0,1,4,0,0,2,1

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

%C From _Jianing Song_, Sep 07 2018: (Start)

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

%C Inverse Moebius transform of A011585. (End)

%C 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

%H G. C. Greubel, <a href="/A035171/b035171.txt">Table of n, a(n) for n = 1..10000</a>

%F From _Jianing Song_, Sep 07 2018: (Start)

%F 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.

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

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

%F (End)

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

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

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

%Y Cf. A028641.

%Y Moebius transform gives A011585.

%Y 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.

%Y 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.

%K nonn,mult

%O 1,5

%A _N. J. A. Sloane_