A084916 Positive numbers of the form k = x^2 - 3*y^2.

1, 4, 6, 9, 13, 16, 22, 24, 25, 33, 36, 37, 46, 49, 52, 54, 61, 64, 69, 73, 78, 81, 88, 94, 96, 97, 100, 109, 117, 118, 121, 132, 141, 142, 144, 148, 150, 157, 166, 169, 177, 181, 184, 193, 196, 198, 208, 213, 214, 216, 222, 225, 229, 241, 244, 249, 253, 256

Offset: 1

Author

Roger Cuculière, Jul 14 2003

Keywords

• nonn

Comments

Equivalently, positive numbers of the form k = x^2 + 2xy - 2y^2. These are equivalent forms, of discriminant 12.

Also numbers representable as x^2 + 4*x*y + y^2 with 0 <= x <= y. - Gheorghe Coserea, Jul 29 2018 [The restriction 0 <= x <= y is not necessary. - Klaus Purath, Feb 05 2023]

From Klaus Purath, Feb 05 2023: (Start)

Also positive numbers of the form x^2 + 2*m*x*y + (m^2 - 3)*y^2. This includes all forms given above so far.

All terms are congruent to {0, 1, 4, 6, 9, 10} modulo 12.

The product of any two terms belongs to the sequence - (empirically secured up to a(k)*a(m) for 2 <= k, m <= 85). Thus it appears that this sequence is closed under multiplication. Perhaps someone can find a proof? (End)

Links

• Jean-François Alcover, Table of n, a(n) for n = 1..1000
• Will Jagy, C++ program Conway_Positive_All.cc to find all positive numbers represented by an indefinite binary quadratic form
• Will Jagy, Sample output from Conway_Positive_All.cc
• Will Jagy, C++ program Conway_Positive_Primitive.cc to find positive numbers primitively represented by an indefinite binary quadratic form
• Will Jagy, Sample output from Conway_Positive_Prim.cc
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Crossrefs

Cf. A031363, A035251, A243655 (primitive representations).

See A068228 for primes.

Programs

Mathematica

Reap[For[n = 1, n < 300, n++, If[Reduce[n == x^2 - 3*y^2, {x, y}, Integers] =!= False, Sow[n]]]][[2, 1]] (* Jean-François Alcover, Dec 03 2013 *)

Extensions

More terms from Reinhard Zumkeller, Jul 17 2003