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A084916
Positive numbers of the form k = x^2 - 3*y^2.
7
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
1,2
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)
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 *)
CROSSREFS
Cf. A031363, A035251, A243655 (primitive representations).
See A068228 for primes.
Sequence in context: A368196 A110607 A085802 * A187910 A283623 A020747
KEYWORD
nonn
AUTHOR
Roger Cuculière, Jul 14 2003
EXTENSIONS
More terms from Reinhard Zumkeller, Jul 17 2003
STATUS
approved