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

1,2

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

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)

Mathematica

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

Prog

(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

Crossrefs

Moebius transform gives A391502.

Cf. A038907 (primes not inert in Q(sqrt(33))), A038908 (prime remaining inert).

Dedekind zeta functions for imaginary quadratic number fields of discriminants D = -3..-47, -67, -163: A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, A035167, A192013, A035159, A035155, A035151, A035180, A035147, A035143, A318982, A318983.

Dedekind zeta functions for real quadratic number fields of discriminants D = 5..41: A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, this sequence, A035219, A035192, A035223.

Sequence in context: A239631 A113288 A199580 * A147654 A373247 A321377

Adjacent sequences: A035212 A035213 A035214 * A035216 A035217 A035218

Keyword

nonn,easy,mult

Author

N. J. A. Sloane

Status

approved