A035179 a(n) = Sum_{d|n} Kronecker(-11, d).

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• mult

Comments

This is a member of an infinite family of odd weight level 11 multiplicative modular forms. g_1 = A035179, g_3 = A129522, g_5 = A065099, g_7 = A138661. - Michael Somos, Jun 07 2015

Half of the number of integer solutions to x^2 + x*y + 3*y^2 = n. - Michael Somos, Jun 05 2005

From Jianing Song, Sep 07 2018: (Start)

Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = -11.

Inverse Moebius transform of A011582. (End)

Coefficients of Dedekind zeta function for the quadratic number field of discriminant -11. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Examples

			G.f. = x + 2*x^3 + x^4 + 2*x^5 + 3*x^9 + x^11 + 2*x^12 + 4*x^15 + x^16 + 2*x^20 + ...
		

References

• Henry McKean and Victor Moll, Elliptic Curves, Cambridge University Press, 1997, page 202. MR1471703 (98g:14032).

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A028609, A065099, A129522, A138661.

Moebius transform gives A011582.

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.

Programs

Magma

A := Basis( ModularForms( Gamma1(11), 1), 88); B := (-1 + A[1] + 2*A[2] + 4*A[4] + 2*A[5]) / 2; B; // Michael Somos, Jun 07 2015

Mathematica

a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -11, #] &]]; (* Michael Somos, Jun 07 2015 *)

PARI

{a(n) = if( n<1, 0, qfrep([2, 1; 1, 6], n, 1)[n])}; \\ Michael Somos, Jun 05 2005

PARI

{a(n) = if( n<1, 0, direuler(p=2, n, 1 / ((1 - X) * (1 - kronecker( -11, p)*X))) [n])}; \\ Michael Somos, Jun 05 2005

PARI

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -11, d)))};

Formula

a(n) is multiplicative with a(11^e) = 1, a(p^e) = (1 + (-1)^e) / 2 if p == 2, 6, 7, 8, 10 (mod 11), a(p^e) = e + 1 if p == 1, 3, 4, 5, 9 (mod 11). - Michael Somos, Jan 29 2007

Moebius transform is period 11 sequence [ 1, -1, 1, 1, 1, -1, -1, -1, 1, -1, 0, ...]. - Michael Somos, Jan 29 2007

G.f.: Sum_{k>0} Kronecker(-11, k) * x^k / (1 - x^k). - Michael Somos, Jan 29 2007

A028609(n) = 2 * a(n) unless n = 0. - Michael Somos, Jun 24 2011

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(11) = 0.947225... . - Amiram Eldar, Oct 11 2022