A035256 Positive integers of the form x^2+3xy-y^2.

1, 3, 4, 9, 12, 13, 16, 17, 23, 25, 27, 29, 36, 39, 43, 48, 49, 51, 52, 53, 61, 64, 68, 69, 75, 79, 81, 87, 92, 100, 101, 103, 107, 108, 113, 116, 117, 121, 127, 129, 131, 139, 144, 147, 153, 156, 157, 159, 169, 172, 173, 179, 181, 183, 191, 192, 196, 199, 204, 207, 208, 211, 212, 221, 225, 233, 237, 243, 244, 251

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

This is an indefinite quadratic form of discriminant 13.

Also, positive integers of the form x^2+6xy-4y^2 (an indefinite quadratic form of discriminant 52).

Also, 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= 13.

From Klaus Purath, May 07 2023: (Start)

Also, positive integers of the form x^2 + (2m+1)xy + (m^2+m-3)y^2, m, x, y integers. This includes the form in the name.

Also, positive integers of the form x^2 + 2mxy + (m^2-13)y^2, m, x, y integers. This includes the form in the comment above.

This sequence contains all squares. The prime factors of the terms except for {2, 5, 7, 11, 19, ...} = A038884 are terms of the sequence. Also the products of terms belong to the sequence. Thus this set of terms is closed under multiplication.

A positive integer N belongs to the sequence if and only if N (modulo 13) is a term of A010376 and, moreover, in the case that prime factors p of N are terms of A038884, they occur only with even exponents. For these prime factors also p (modulo 13) = {2, 5, 6, 7, 8, 11} applies. (End)

Links

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

Crossrefs

Primes in this sequence = A038883 and A141188.

Cf. A035195.

Programs

Mathematica

formQ[n_] := Reduce[a > 0 && b > 0 && n == a^2 + 3 a*b - b^2, {a, b}, Integers] =!= False; Select[Range[100], formQ] (* Wesley Ivan Hurt, Jun 18 2014 *)

PARI

m=13; select(x -> x, direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X)), 1) \\ Fixed by Andrey Zabolotskiy, Jul 30 2020

Extensions

Entry revised by N. J. A. Sloane, Jun 01 2014