A035203 - OEIS

%I #19 Nov 19 2023 01:29:05

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

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

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

%N Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 21.

%C Coefficients of Dedekind zeta function for the quadratic number field of discriminant 21. See A002324 for formula and Maple code. - _N. J. A. Sloane_, Mar 22 2022

%H Amiram Eldar, <a href="/A035203/b035203.txt">Table of n, a(n) for n = 1..10000</a>

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log((5+sqrt(21))/2)/sqrt(21) = 0.683807... . - _Amiram Eldar_, Oct 11 2022

%F From _Amiram Eldar_, Nov 19 2023: (Start)

%F a(n) = Sum_{d|n} Kronecker(21, d).

%F Multiplicative with a(p^e) = 1 if Kronecker(21, p) = 0 (p = 3 or 7), a(p^e) = (1+(-1)^e)/2 if Kronecker(21, p) = -1 (p is in A038894), and a(p^e) = e+1 if Kronecker(21, p) = 1 (p is in A038893 \ {3, 7}). (End)

%t a[n_] := DivisorSum[n, KroneckerSymbol[21, #] &]; Array[a, 100] (* _Amiram Eldar_, Oct 11 2022 *)

%o (PARI) my(m = 21); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

%o (PARI) a(n) = sumdiv(n, d, kronecker(21, d)); \\ _Amiram Eldar_, Nov 19 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. A038893, A038894.

%K nonn,easy,mult

%O 1,5

%A _N. J. A. Sloane_