A035182 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = -7.

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

Offset

1,2

Comments

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + 5*v^2 + 4*w^2 - 8*v*w - 4*u*v + 2*u*w + v - w. - Michael Somos, Jul 21 2004

Half of the number of integer solutions to x^2 + x*y + 2*y^2 = n. - Michael Somos, Jun 05 2005

Inverse Moebius transform of A175629. - Jianing Song, Sep 07 2018

Coefficients of Dedekind zeta function for the quadratic number field of discriminant -7. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Links

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

Formula

a(n) is multiplicative with a(7^e) = 1, a(p^e) = e + 1 if p == 1, 2, 4 (mod 7), a(p^e) = (1 + (-1)^e) / 2 if p == 3, 5, 6 (mod 7). - Michael Somos, May 28 2005

2 * a(n) = A002652(n) unless n = 0.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(7) = 1.187410... (A326919). - Amiram Eldar, Oct 11 2022

Example

G.f. = x + 2*x^2 + 3*x^4 + x^7 + 4*x^8 + x^9 + 2*x^11 + 2*x^14 + 5*x^16 + ...

Mathematica

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -7, d], { d, Divisors[ n]}]]; (* Michael Somos, Jan 23 2014 *)
a[ n_] := If[ n < 1, 0, Length @ FindInstance[ n == x^2 + x y + 2 y^2, {x, y}, Integers, 10^9] / 2]; (* Michael Somos, Jan 23 2014 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -7, #] &]]; (* Michael Somos, Jun 10 2015 *)

Prog

(PARI) {a(n) = my(A, p, e); if( n<0, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; [ !(e%2), 1, e+1] [kronecker( -7, p) + 2]))}; \\ Michael Somos, May 28 2005
(PARI) {a(n) = if( n<1, 0, qfrep([ 2, 1; 1, 4], n, 1)[n])}; \\ Michael Somos, Jun 05 2005
(PARI) {a(n) = if( n<1, 0, direuler( p=2, n, 1 / ((1 - X) * (1 - kronecker( -7, p)*X)))[n])}; \\ Michael Somos, Jun 05 2005
(Magma) A := Basis( ModularForms( Gamma1(14), 1), 106); B<q> := (-1 + A[1] + 2*A[2] + 4*A[3] + 6*A[5]) / 2; B; // Michael Somos, Jun 10 2015

Crossrefs

Cf. A002652, A326919.

Moebius transform gives A175629.

Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.

Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.

Sequence in context: A109362 A085246 A268726 * A280720 A245964 A141700

Adjacent sequences: A035179 A035180 A035181 * A035183 A035184 A035185

Keyword

nonn,mult

Author

N. J. A. Sloane

Status

approved