A035147 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = -43.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• mult

Comments

From Jianing Song, Sep 07 2018: (Start)

Half of the number of integer solutions to x^2 + x*y + 11*y^2 = n.

Inverse Moebius transform of A011591. (End)

Links

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

Crossrefs

Cf. A138811.

Moebius transform gives A011591.

Programs

Mathematica

a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-43, #] &]];
Table[a[n], {n, 1, 100}] (* G. C. Greubel, Apr 25 2018 *)

PARI

my(m=-43); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

PARI

a(n) = sumdiv(n, d, kronecker(-43, d)); \\ Amiram Eldar, Nov 18 2023

Formula

From Jianing Song, Sep 07 2018: (Start)

a(n) is multiplicative with a(43^e) = 1, a(p^e) = (1 + (-1)^e) / 2 if Kronecker(-43, p) = -1, a(p^e) = e + 1 if Kronecker(-43, p) = 1.

G.f.: Sum_{k>0} Kronecker(-43, k) * x^k / (1 - x^k).

A138811(n) = 2 * a(n) unless n = 0. (End)

From Amiram Eldar, Nov 18 2023: (Start)

a(n) = Sum_{d|n} Kronecker(-43, d).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(43) = 0.479088... . (End)