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A035219 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 37.

%I A035219 #11 Nov 20 2023 11:53:18
%S A035219 1,0,2,1,0,0,2,0,3,0,2,2,0,0,0,1,0,0,0,0,4,0,0,0,1,0,4,2,0,0,0,0,4,0,
%T A035219 0,3,1,0,0,0,2,0,0,2,0,0,2,2,3,0,0,0,2,0,0,0,0,0,0,0,0,0,6,1,0,0,2,0,
%U A035219 0,0,2,0,2,0,2,0,4,0,0,0,5
%N A035219 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 37.
%C A035219 Coefficients of Dedekind zeta function for the quadratic number field of discriminant 37. See A002324 for formula and Maple code. - _N. J. A. Sloane_, Mar 22 2022
%H A035219 Amiram Eldar, <a href="/A035219/b035219.txt">Table of n, a(n) for n = 1..10000</a>
%F A035219 From _Amiram Eldar_, Nov 20 2023: (Start)
%F A035219 a(n) = Sum_{d|n} Kronecker(37, d).
%F A035219 Multiplicative with a(37^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(37, p) = -1 (p is in A038914), and a(p^e) = e+1 if Kronecker(37, p) = 1 (p is in A191027).
%F A035219 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(sqrt(37)+6)/sqrt(37) = 0.819292168725... . (End)
%t A035219 a[n_] := DivisorSum[n, KroneckerSymbol[37, #] &]; Array[a, 100] (* _Amiram Eldar_, Nov 20 2023 *)
%o A035219 (PARI) my(m = 37); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
%o A035219 (PARI) a(n) = sumdiv(n, d, kronecker(37, d)); \\ _Amiram Eldar_, Nov 20 2023
%Y A035219 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 A035219 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.
%Y A035219 Cf. A038914, A191027.
%K A035219 nonn,easy,mult
%O A035219 1,3
%A A035219 _N. J. A. Sloane_