A035188 - OEIS

%I #18 Nov 20 2023 11:53:08

%S 1,1,1,1,2,1,0,1,1,2,0,1,0,0,2,1,0,1,2,2,0,0,2,1,3,0,1,0,2,2,0,1,0,0,

%T 0,1,0,2,0,2,0,0,2,0,2,2,2,1,1,3,0,0,2,1,0,0,2,2,0,2,0,0,0,1,0,0,2,0,

%U 2,0,2,1,2,0,3,2,0,0,0,2,1

%N Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 6.

%C Coefficients of Dedekind zeta function for the quadratic number field of discriminant 24. See A002324 for formula and Maple code. - _N. J. A. Sloane_, Mar 22 2022

%H G. C. Greubel, <a href="/A035188/b035188.txt">Table of n, a(n) for n = 1..10000</a>

%F From _Amiram Eldar_, Oct 17 2022: (Start)

%F a(n) = Sum_{d|n} Kronecker(6, d).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(5+2*sqrt(6)) / sqrt(6) = 0.935881... . (End)

%F Multiplicative with a(p^e) = 1 if Kronecker(6, p) = 0 (p = 2 or 3), a(p^e) = (1+(-1)^e)/2 if Kronecker(6, p) = -1 (p is in A038877), and a(p^e) = e+1 if Kronecker(6, p) = 1 (p is in A097934). - _Amiram Eldar_, Nov 20 2023

%t a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[6, #] &]]; Table[ a[n], {n, 1, 100}] (* _G. C. Greubel_, Apr 27 2018 *)

%o (PARI) my(m=6); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

%o (PARI) a(n) = sumdiv(n, d, kronecker(6, d)); \\ _Amiram Eldar_, Nov 20 2023

%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.

%Y Cf. A038877, A097934.

%K nonn,easy,mult

%O 1,5

%A _N. J. A. Sloane_