A139512 Primes of the form x^2 + 32*x*y + y^2 for x and y nonnegative.

229, 349, 409, 421, 661, 769, 829, 1021, 1069, 1249, 1381, 1429, 1549, 1789, 1801, 1861, 2089, 2161, 2269, 2389, 3001, 3061, 3109, 3181, 3229, 3469, 3889, 4021, 4129, 4201, 4441, 4861, 4909, 5101, 5449, 5521, 5869, 5881, 6121, 6469, 6481, 6529, 6781

Offset

1,1

Comments

Are all terms == 1 mod 12? - Zak Seidov, Apr 25 2008

Yes: (i) all terms == 1 mod 3 because the quadratic form has terms == {0,1} mod 3 and the values ==0 mod 3 are not primes. (ii) all terms == 1 mod 4 because the quadratic form has terms == {0,1,2} mod 4 and the values = {0,2} mod 4 are not primes. By the Chinese remainder constructions for coprime 3 and 4 all prime terms are == 1 mod 12. - R. J. Mathar, Jun 10 2020

Links

Robert Israel, Table of n, a(n) for n = 1..10000

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references), discriminant 1020.

Maple

N:= 10^4: # for terms <= N
R:= {}:
for x from 0 by 2 while x^2 <= N do
   R:= R union select(isprime, {seq(x^2 + 32*x*y + y^2, y = 1 .. floor(sqrt(255*x^2+N))-16*x, 2)})
od:
sort(convert(R, list)); # Robert Israel, May 31 2026

Mathematica

a = {}; w = 32; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a] (*Artur Jasinski*)

Crossrefs

Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

Sequence in context: A157348 A142221 A142779 * A142263 A142608 A210514

Adjacent sequences: A139509 A139510 A139511 * A139513 A139514 A139515

Keyword

nonn

Author

Artur Jasinski, Apr 24 2008

Status

approved