A035215 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 33.

1, 2, 1, 3, 0, 2, 0, 4, 1, 0, 1, 3, 0, 0, 0, 5, 2, 2, 0, 0, 0, 2, 0, 4, 1, 0, 1, 0, 2, 0, 2, 6, 1, 4, 0, 3, 2, 0, 0, 0, 2, 0, 0, 3, 0, 0, 0, 5, 1, 2, 2, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 4, 0, 7, 0, 2, 2, 6, 0, 0, 0, 4, 0, 4, 1, 0, 0, 0, 0, 0, 1

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• mult

Comments

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 33. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

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.

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.

Cf. A038907, A038908.

Programs

Mathematica

a[n_] := DivisorSum[n, KroneckerSymbol[33, #] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2023 *)

PARI

my(m = 33); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

PARI

a(n) = sumdiv(n, d, kronecker(33, d)); \\ Amiram Eldar, Nov 19 2023

Formula

From Amiram Eldar, Nov 19 2023: (Start)

a(n) = Sum_{d|n} Kronecker(33, d).

Multiplicative with a(p^e) = 1 if Kronecker(33, p) = 0 (p = 3 or 11), a(p^e) = (1+(-1)^e)/2 if Kronecker(33, p) = -1 (p is in A038908), and a(p^e) = e+1 if Kronecker(33, p) = 1 (p is in A038907 \ {3, 11}).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(4*sqrt(33)+23)/sqrt(33) = 1.332797188186... . (End)