A035151 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -39.

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

Offset

1,2

Links

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

Formula

From Amiram Eldar, Nov 17 2023: (Start)

a(n) = Sum_{d|n} Kronecker(-39, d).

Multiplicative with a(p^e) = 1 if Kronecker(-39, p) = 0 (p = 3 or 13), a(p^e) = (1+(-1)^e)/2 if Kronecker(-39, p) = -1 (p is in A191070), and a(p^e) = e+1 if Kronecker(-39, p) = 1 (p is in A191029).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*Pi/sqrt(39) = 2.0122297... . (End)

Mathematica

a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-39, #] &]];
Table[a[n], {n, 1, 100}] (* G. C. Greubel, Apr 25 2018 *)

Prog

(PARI) my(m=-39); direuler(p=2, 101, 1/(1-(kronecker(m, p)*(X-X^2))-X))
(PARI) a(n) = sumdiv(n, d, kronecker(-39, d)); \\ Amiram Eldar, Nov 17 2023

Crossrefs

Cf. A191029, A191070.

Sequence in context: A021473 A266756 A035181 * A290536 A352570 A277855

Adjacent sequences: A035148 A035149 A035150 * A035152 A035153 A035154

Keyword

nonn,easy,mult

Author

N. J. A. Sloane

Status

approved