A035179 - OEIS

%I #41 Oct 11 2022 06:17:25

%S 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,

%T 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,

%U 4,0,2,0,0,0,6,0,0,0,0,2,5,0,0,0,0,0,0

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

%C 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

%C Half of the number of integer solutions to x^2 + x*y + 3*y^2 = n. - _Michael Somos_, Jun 05 2005

%C From _Jianing Song_, Sep 07 2018: (Start)

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

%C Inverse Moebius transform of A011582. (End)

%C 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

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

%H G. C. Greubel, <a href="/A035179/b035179.txt">Table of n, a(n) for n = 1..5000</a>

%F 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

%F Moebius transform is period 11 sequence [ 1, -1, 1, 1, 1, -1, -1, -1, 1, -1, 0, ...]. - _Michael Somos_, Jan 29 2007

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

%F A028609(n) = 2 * a(n) unless n = 0. - _Michael Somos_, Jun 24 2011

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

%e 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 + ...

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

%o (PARI) {a(n) = if( n<1, 0, qfrep([2, 1; 1, 6], 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( -11, p)*X))) [n])}; \\ _Michael Somos_, Jun 05 2005

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

%o (Magma) A := Basis( ModularForms( Gamma1(11), 1), 88); B<q> := (-1 + A[1] + 2*A[2] + 4*A[4] + 2*A[5]) / 2; B; // _Michael Somos_, Jun 07 2015

%Y Cf. A028609, A065099, A129522, A138661.

%Y Moebius transform gives A011582.

%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 1,3

%A _N. J. A. Sloane_