A028954 -id:A028954 - cp's OEIS frontend

A028609 Expansion of (theta_3(z)theta_3(11z) + theta_2(z)theta_2(11z)).

1, 2, 0, 4, 2, 4, 0, 0, 0, 6, 0, 2, 4, 0, 0, 8, 2, 0, 0, 0, 4, 0, 0, 4, 0, 6, 0, 8, 0, 0, 0, 4, 0, 4, 0, 0, 6, 4, 0, 0, 0, 0, 0, 0, 2, 12, 0, 4, 4, 2, 0, 0, 0, 4, 0, 4, 0, 0, 0, 4, 8, 0, 0, 0, 2, 0, 0, 4, 0, 8, 0, 4, 0, 0, 0, 12, 0, 0, 0, 0, 4, 10, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 8, 0, 0, 0, 4, 0, 6, 6, 0, 0

Offset: 0

N. J. A. Sloane

Theta series of lattice with Gram matrix [2, 1; 1, 6].
Number of integer solutions (x, y) to x^2 + x*y + 3*y^2 = n. - Michael Somos, Sep 20 2004
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + 2*x + 4*x^3 + 2*x^4 + 4*x^5 + 6*x^9 + 2*x^11 + 4*x^12 + 8*x^15 + ...
Theta series of lattice with Gram matrix [2, 1; 1, 6] = 1 + 2*q^2 + 4*q^6 + 2*q^8 + 4*q^10 + 6*q^18 + 2*q^22 + 4*q^24 + 8*q^30 + 2*q^32 + 4*q^40 + 4*q^46 + 6*q^50 + 8*q^54 + 4*q^62 + 4*q^66 + 6*q^72 + 4*q^74 + ...

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

John Cannon, Table of n, a(n) for n = 0..5000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A028610, A035179, A028954, A056874.
Cf. A000122, A000700, A010054, A121373.
Number of integer solutions to f(x,y) = n where f(x,y) is the principal binary quadratic form with discriminant d: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), A033715 (d=-8), this sequence (d=-11), A028641 (d=-19), A138811 (d=-43).

A := Basis( ModularForms( Gamma1(11), 1), 103); A[1] + 2*A[2] + 4*A[4] + 2*A[5]; /* Michael Somos, Jul 12 2014 */

a[ n_] := If[ n < 1, Boole[ n == 0], DivisorSum[ n, KroneckerSymbol[ -11, #] &] 2]; (* Michael Somos, Jul 12 2014 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^11] + EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^11], {q, 0, n}]; (* Michael Somos, Jul 12 2014 *)

{a(n) = my(t); if( n<1, n==0, 2 * issquare(n) + 2 * sum( y=1, sqrtint(n * 4\11), 2 * issquare( t=4*n - 11*y^2) - (t==0)))}; /* Michael Somos, Sep 20 2004 */

{a(n) = if( n<0, 0, polcoeff( 1 + 2 * x * Ser(qfrep( [ 2, 1; 1, 6], n, 1)), n))}; /* Michael Somos, Apr 21 2015 */

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

{a(n) = if( n<1, n==0, 2 * sumdiv( n, d, kronecker( -11, d)))}; /* Michael Somos, Jan 29 2007 */

Expansion of phi(x) * phi(x^11) = 4 * x^3 * psi(x^2) * psi(x^22) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Apr 21 2015
From Michael Somos, Jan 29 2007: (Start)
Moebius transform is period 11 sequence [ 2, -2, 2, 2, 2, -2, -2, -2, 2, -2, 0, ...].
a(n) = 2 * b(n) and b(n) is multiplicative with b(11^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if p == 2, 6, 7, 8, 10 (mod 11), b(p^e) = e + 1 if p == 1, 3, 4, 5, 9 (mod 11).
G.f.: 1 + 2 * Sum_{k>0} Kronecker( -11, k) * x^k / (1 - x^k). (End)
G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 05 2007
Expansion of (F(x)^2 + 4 * F(x^2)^2 + 8 * F(x^4)^2) / F(x^2) in powers of x or expansion of (F(x)^2 + 2 * F(x^2)^2 + 2 * F(x^4)^2) / F(x^2) in powers of x^4 where F(x) = x^(1/2) * f(-x) * f(-x^11) and f() is a Ramanujan theta function. - Michael Somos, Mar 01 2010
a(n) = 2 * A035179(n) unless n=0. Convolution square is A028610.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 2*Pi/sqrt(11) = 1.894451... . - Amiram Eldar, Dec 16 2023

A056874 Primes of form x^2+xy+3y^2, discriminant -11.

Original entry on oeis.org

3, 5, 11, 23, 31, 37, 47, 53, 59, 67, 71, 89, 97, 103, 113, 137, 157, 163, 179, 181, 191, 199, 223, 229, 251, 257, 269, 311, 313, 317, 331, 353, 367, 379, 383, 389, 397, 401, 419, 421, 433, 443, 449, 463, 467, 487, 499, 509, 521, 577, 587, 599

Offset: 1

N. J. A. Sloane, Sep 02 2000

Also, primes of form (x^2+11*y^2)/4.
Also, primes of the form x^2-xy+3y^2 with x and y nonnegative. - T. D. Noe, May 07 2005
Primes congruent to 0, 1, 3, 4, 5 or 9 (mod 11). As this discriminant has class number 1, all binary quadratic forms ax^2+bxy+cy^2 with b^2-4ac=-11 represent these primes. - Rick L. Shepherd, Jul 25 2014
Also, primes which are squares (mod 11) (or, (mod 22), cf. A191020). - M. F. Hasler, Jan 15 2016
Also, primes p such that Legendre(-11,p) = 0 or 1. - N. J. A. Sloane, Dec 25 2017

Vincenzo Librandi, N. J. A. Sloane and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi, next 4000 terms from N. J. A. Sloane]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A002346 and A002347 for values of x and y.

Primes in A028954.

QuadPrimes2[1, 1, 3, 100000] (* see A106856 *)

{ fc2(a,b,c,M) = my(p,t1,t2,n);
m = 0;
for(n=1,M, p = prime(n);
t2 = qfbsolve(Qfb(a,b,c),p); if(t2 == 0,, m++; print(m," ",p )));
}
fc2(1,-1,3,10703);

Edited by N. J. A. Sloane, Jun 01 2014 and Jun 16 2014

A106857 Primes of the form x^2+xy+3y^2, with x and y nonnegative.

Original entry on oeis.org

3, 5, 23, 31, 37, 47, 53, 59, 67, 89, 97, 113, 157, 163, 179, 181, 191, 199, 251, 257, 269, 311, 313, 317, 331, 367, 379, 383, 389, 401, 419, 433, 443, 449, 463, 487, 499, 509, 521, 577, 587, 599, 617, 641, 643, 647, 653, 683, 691, 709, 719, 727, 751, 757

Offset: 1

T. D. Noe, May 09 2005

Discriminant=-11.

N. J. A. Sloane and Vincenzo Librarndi, Table of n, a(n) for n = 1..10000 [First 3000 terms from Vincenzo Librarndi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

A056874 is the main sequence for these primes. Cf. A028954.

QuadPrimes2[a_, b_, c_, lmt_] := Module[{p, d, lst = {}, xMax, yMax}, d = b^2 - 4a*c; If[a > 0 && c > 0 && d < 0, xMax = Sqrt[lmt/a]*(1+Abs[b]/Floor[Sqrt[-d]])]; Do[ If[ 4c*lmt + d*x^2 >= 0, yMax = ((-b)*x + Sqrt[4c*lmt + d*x^2])/(2c), yMax = 0 ]; Do[p = a*x^2 + b*x*y + c*y^2; If[ PrimeQ[ p]  && !MemberQ[ lst, p], AppendTo[ lst, p]], {y, 0, yMax}], {x, 0, xMax}]; Sort[ lst]];
t2 = QuadPrimes2[1, 1, 3, 1000]

Replaced defective Mma program, extended b-file. - N. J. A. Sloane, Jun 16 2014

A035247 Indices of the nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m= -11.

Original entry on oeis.org

1, 3, 4, 5, 9, 11, 12, 15, 16, 20, 23, 25, 27, 31, 33, 36, 37, 44, 45, 47, 48, 49, 53, 55, 59, 60, 64, 67, 69, 71, 75, 80, 81, 89, 92, 93, 97, 99, 100, 103, 108, 111, 113, 115, 121, 124, 125, 132, 135, 137, 141, 144, 147, 148, 155, 157, 159, 163, 165, 169, 176, 177, 179, 180, 181, 185, 188, 191, 192, 196, 199

Offset: 1

N. J. A. Sloane

nonn

Cf. A028954 (a probable duplicate). [From R. J. Mathar, Oct 20 2008]

Cf. A035179 (the expansion itself).

Reap[For[n = 1, n < 200, n++, r = Reduce[x^2 + x y + 3 y^2 == n, {x, y}, Integers]; If[r =!= False, Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 31 2016 *)

m=-11; select(x -> x, direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X)), 1) \\ Fixed by Andrey Zabolotskiy, Jul 30 2020

More terms from Jean-François Alcover, Oct 31 2016

Name corrected by Andrey Zabolotskiy, Jul 30 2020

cp's OEIS Frontend

A028609 Expansion of (theta_3(z)*theta_3(11z) + theta_2(z)*theta_2(11z)).

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A056874 Primes of form x^2+xy+3y^2, discriminant -11.

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A106857 Primes of the form x^2+xy+3y^2, with x and y nonnegative.

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A035247 Indices of the nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m= -11.

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A028609 Expansion of (theta_3(z)theta_3(11z) + theta_2(z)theta_2(11z)).

A035247 Indices of the nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m= -11.