A035213 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 31.

1, 2, 2, 3, 2, 4, 0, 4, 3, 4, 2, 6, 0, 0, 4, 5, 0, 6, 0, 6, 0, 4, 2, 8, 3, 0, 4, 0, 0, 8, 1, 6, 4, 0, 0, 9, 0, 0, 0, 8, 2, 0, 2, 6, 6, 4, 0, 10, 1, 6, 0, 0, 0, 8, 4, 0, 0, 0, 0, 12, 0, 2, 0, 7, 0, 8, 0, 0, 4, 0, 0, 12, 0, 0, 6, 0, 0, 0, 2, 10

Offset

1,2

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(31, d).

Multiplicative with a(31^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(31, p) = -1 (p is in A038906), and a(p^e) = e+1 if Kronecker(31, p) = 1 (p is in A038905 \ {31}).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(273*sqrt(31)+1520)/sqrt(31) = 2.880729917283... . (End)

Mathematica

a[n_] := DivisorSum[n, KroneckerSymbol[31, #] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2023 *)

Prog

(PARI) my(m = 31); direuler(p=2, 101, 1/(1-(kronecker(m, p)*(X-X^2))-X))
(PARI) a(n) = sumdiv(n, d, kronecker(31, d)); \\ Amiram Eldar, Nov 19 2023

Crossrefs

Cf. A038905, A038906.

Sequence in context: A117122 A122828 A242213 * A340224 A083901 A274517

Adjacent sequences: A035210 A035211 A035212 * A035214 A035215 A035216

Keyword

nonn,easy,mult

Author

N. J. A. Sloane

Status

approved