A318982 - OEIS

A318982

a(n) = Sum_{d|n} Kronecker(-67, d).

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

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OFFSET

1,17

COMMENTS

Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = -67.

Half of the number of integer solutions to x^2 + x*y + 17*y^2 = n. Also, a(n) is the number of integral elements with norm n in Q[sqrt(-67)] counted up to association.

Inverse Moebius transform of A011596.

LINKS

Jianing Song, Table of n, a(n) for n = 1..10000

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS.

FORMULA

a(n) is multiplicative with a(67^e) = 1, a(p^e) = (1 + (-1)^e) / 2 if Kronecker(-67, p) = -1, a(p^e) = e + 1 if Kronecker(-67, p) = 1.

G.f.: Sum_{k>0} Kronecker(-67, k) * x^k / (1 - x^k).

A318984(n) = 2 * a(n) unless n = 0.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(67) = 0.383806... . - Amiram Eldar, Dec 16 2023

EXAMPLE

G.f. = x + x^4 + x^9 + x^16 + 2*x^17 + 2*x^19 + 2*x^23 + x^25 + 2*x^29 + x^36 + 2*x^37 + 2*x^47 + x^49 + 2*x^59 + x^64 + x^67 + 2*x^68 + 2*x^71 + 2*x^73 + 2*x^76 + ...

MATHEMATICA

a[n_]:=If[n<0, 0, DivisorSum[n, KroneckerSymbol[-67, #] &]];

Table[a[n], {n, 1, 110}] (* Vincenzo Librandi, Sep 10 2018 *)

PROG

(PARI) a(n) = sumdiv(n, d, kronecker(-67, d))

CROSSREFS

Cf. A318984.

Moebius transform gives A011596.

Cf. A106933 (primes not inert in Q(sqrt(-67))), A191041 (primes decomposing), A191077 (primes remaining inert).