A020674 -id:A020674 - cp's OEIS frontend

A106889 Primes of the form 2x^2 + 5y^2.

2, 5, 7, 13, 23, 37, 47, 53, 103, 127, 157, 167, 173, 197, 223, 263, 277, 293, 317, 367, 373, 383, 397, 463, 487, 503, 557, 607, 613, 647, 653, 677, 727, 733, 743, 757, 773, 797, 823, 853, 863, 877, 887, 967, 983, 997, 1013, 1063, 1087, 1093, 1103, 1117

Offset: 1

T. D. Noe, May 09 2005

Discriminant = -40.

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

Cf. A139827. Primes in A020674.

QuadPrimes2[2, 0, 5, 10000] (* see A106856 *)

list(lim)=my(v=List(),w,t); for(x=0, sqrtint(lim\2), w=2*x^2; for(y=0, sqrtint((lim-w)\5), if(isprime(t=w+5*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 09 2017

The primes are congruent to {2, 5, 7, 13, 23, 37} (mod 40). - T. D. Noe, May 02 2008

A317642 Expansion of theta_3(q^2)*theta_3(q^5), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 0, 2, 0, 0, 2, 0, 4, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 2, 0, 4, 4, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 2, 0, 4, 0, 0, 2, 0, 4, 4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 8, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 4, 0, 0, 6

Offset: 0

Ilya Gutkovskiy, Aug 02 2018

nonn

Number of integer solutions to the equation 2*x^2 + 5*y^2 = n.

			G.f. = 1 + 2*q^2 + 2*q^5 + 4*q^7 + 2*q^8 + 4*q^13 + 2*q^18 + 2*q^20 + 4*q^22 + ...

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Cf. A000286, A020674, A033718, A106889, A108563, A192323.

nmax = 98; CoefficientList[Series[EllipticTheta[3, 0, q^2] EllipticTheta[3, 0, q^5], {q, 0, nmax}], q]
nmax = 98; CoefficientList[Series[QPochhammer[-q^2, -q^2] QPochhammer[-q^5, -q^5]/(QPochhammer[q^2, -q^2] QPochhammer[q^5, -q^5]), {q, 0, nmax}], q]

G.f.: Product_{k>=1} (1 + x^(4*k-2))^2*(1 - x^(4*k))*(1 + x^(10*k-5))^2*(1 - x^(10*k)).

cp's OEIS Frontend

A106889 Primes of the form 2x^2 + 5y^2.

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