A035210 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 28.

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

Offset

1,3

Comments

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 28. 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(28, d).

Multiplicative with a(p^e) = 1 if Kronecker(28, p) = 0 (p = 2 or 7), a(p^e) = (1+(-1)^e)/2 if Kronecker(28, p) = -1 (p is in A003632), and a(p^e) = e+1 if Kronecker(28, p) = 1 (p is in A296934).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(3*sqrt(7)+8)/sqrt(7) = 1.046454884756... . (End)

Mathematica

a[n_] := DivisorSum[n, KroneckerSymbol[28, #] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2023 *)

Prog

(PARI) my(m = 28); direuler(p=2, 101, 1/(1-(kronecker(m, p)*(X-X^2))-X))
(PARI) a(n) = sumdiv(n, d, kronecker(28, d)); \\ Amiram Eldar, Nov 19 2023

Crossrefs

Moebius transform gives A390615.

Cf. A038878 (primes not inert in Q(sqrt(7))), A296934 (primes decomposing), A003632 (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, this sequence, A035211, A035215, A035219, A035192, A035223.

Sequence in context: A241272 A358991 A033780 * A366750 A185207 A185206

Adjacent sequences: A035207 A035208 A035209 * A035211 A035212 A035213

Keyword

nonn,easy,mult

Author

N. J. A. Sloane

Status

approved