This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A035171 #24 Oct 11 2022 06:16:06
%S A035171 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 A035171 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 A035171 0,0,0,0,2,0,0,1,4,0,0,2,1
%N 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.
%C A035171 From _Jianing Song_, Sep 07 2018: (Start)
%C A035171 Half of the number of integer solutions to x^2 + x*y + 5*y^2 = n.
%C A035171 Inverse Moebius transform of A011585. (End)
%C A035171 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 A035171 G. C. Greubel, <a href="/A035171/b035171.txt">Table of n, a(n) for n = 1..10000</a>
%F A035171 From _Jianing Song_, Sep 07 2018: (Start)
%F A035171 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 A035171 G.f.: Sum_{k>0} Kronecker(-19, k) * x^k / (1 - x^k).
%F A035171 A028641(n) = 2 * a(n) unless n = 0.
%F A035171 (End)
%F A035171 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(19) = 0.720730... . - _Amiram Eldar_, Oct 11 2022
%t A035171 a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-19, #] &]]; Table[a[n], {n, 1, 100}] (* _G. C. Greubel_, Jul 17 2018 *)
%o A035171 (PARI) m = -19; direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
%Y A035171 Cf. A028641.
%Y A035171 Moebius transform gives A011585.
%Y A035171 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 A035171 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 A035171 nonn,mult
%O A035171 1,5
%A A035171 _N. J. A. Sloane_