A035188 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 6.

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, 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, 2, 0, 2, 1, 2, 0, 3, 2, 0, 0, 0, 2, 1

Offset

1,5

Comments

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

Links

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

Formula

From Amiram Eldar, Oct 17 2022: (Start)

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

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

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

Mathematica

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

Prog

(PARI) my(m=6); direuler(p=2, 101, 1/(1-(kronecker(m, p)*(X-X^2))-X))
(PARI) a(n) = sumdiv(n, d, kronecker(6, d)); \\ Amiram Eldar, Nov 20 2023

Crossrefs

Moebius transform gives A322796.

Cf. A038876 (primes not inert in Q(sqrt(6))), A097934 (primes decomposing), A038877 (primes 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, this sequence, A035210, A035211, A035215, A035219, A035192, A035223.

Sequence in context: A336727 A316101 A211452 * A342148 A066295 A132004

Adjacent sequences: A035185 A035186 A035187 * A035189 A035190 A035191

Keyword

nonn,easy,mult

Author

N. J. A. Sloane

Status

approved