OFFSET
1,11
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
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)
MATHEMATICA
a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-43, #] &]];
Table[a[n], {n, 1, 100}] (* G. C. Greubel, Apr 25 2018 *)
PROG
(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
CROSSREFS
Cf. A138811.
Moebius transform gives A011591.
Cf. A106891 (primes not inert in Q(sqrt(-43))), A191031 (primes decomposing), A184902 (primes 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, this sequence, A035143, A318982, A318983.
KEYWORD
nonn,easy,mult
AUTHOR
STATUS
approved
