A035195 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 13.

1, 0, 2, 1, 0, 0, 0, 0, 3, 0, 0, 2, 1, 0, 0, 1, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 4, 0, 2, 0, 0, 0, 0, 0, 0, 3, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 1, 0, 4, 1, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 5

Offset

1,3

Comments

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 13. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Formula

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log((3+sqrt(13))/2)/sqrt(13) = 0.662735... . - Amiram Eldar, Oct 11 2022

From Amiram Eldar, Nov 18 2023: (Start)

a(n) = Sum_{d|n} Kronecker(13, d).

Multiplicative with a(13^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(13, p) = -1 (p is in A038884), and a(p^e) = e+1 if Kronecker(13, p) = 1 (p is in A038883 \ {13}). (End)

Mathematica

a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[13, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)

Prog

(PARI) my(m=13); direuler(p=2, 101, 1/(1-(kronecker(m, p)*(X-X^2))-X))
(PARI) a(n) = sumdiv(n, d, kronecker(13, d)); \\ Amiram Eldar, Nov 18 2023

Crossrefs

Moebius transform gives A011583.

Cf. A038883 (primes not inert in Q(sqrt(13))), A296937 (primes decomposing), A038884 (primes remaining inert), A120330 (primes not decomposing).

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, this sequence, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, A035223.

Sequence in context: A331812 A368821 A250002 * A073797 A037856 A037874

Adjacent sequences: A035192 A035193 A035194 * A035196 A035197 A035198

Keyword

nonn,easy,mult

Author

N. J. A. Sloane

Status

approved