A035182 - OEIS

%I #38 Oct 11 2022 06:18:15

%S 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,

%T 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,

%U 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

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

%C 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

%C Half of the number of integer solutions to x^2 + x*y + 2*y^2 = n. - _Michael Somos_, Jun 05 2005

%C Inverse Moebius transform of A175629. - _Jianing Song_, Sep 07 2018

%C 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

%H G. C. Greubel, <a href="/A035182/b035182.txt">Table of n, a(n) for n = 1..10000</a>

%F 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

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

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

%e 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 + ...

%t a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -7, d], { d, Divisors[ n]}]]; (* _Michael Somos_, Jan 23 2014 *)

%t 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 *)

%t a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -7, #] &]]; (* _Michael Somos_, Jun 10 2015 *)

%o (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

%o (PARI) {a(n) = if( n<1, 0, qfrep([ 2, 1; 1, 4], n, 1)[n])}; \\ _Michael Somos_, Jun 05 2005

%o (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

%o (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

%Y Cf. A002652, A326919.

%Y Moebius transform gives A175629.

%Y 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.

%Y 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.

%K nonn,mult

%O 1,2

%A _N. J. A. Sloane_