A139490 Numbers n such that the quadratic form x^2 + n*x*y + y^2 represents exactly the same primes as the quadratic form x^2 + m*y^2 for some m.

1, 4, 6, 7, 8, 10, 14, 16, 18, 22, 26, 38, 58, 82, 86

Offset

1,2

Comments

For the numbers m see A139491.

Conjecture: This sequence is finite and complete (checked for range n<=200 and m<=500).

Three more terms were found by searching n <= 1000 and m <= 4000. The corresponding m are 840, 840, and 1848, which are idoneal numbers A000926. The sequence is probably complete now. [T. D. Noe, Apr 27 2009]

Links

Table of n, a(n) for n=1..15.

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

Example

a(1)=1 because the primes represented by x^2+xy+y^2 are the same as the primes represented by x^2 + 3*y^2 (see A007645).
The known pairs (n,m) are the following (checked for range n<=200 and m<=500):
n={1, 4, 4, 6, 6, 7, 8, 8, 10, 10, 10, 14, 14, 14, 16, 18, 22, 22, 26, 38, 38}
m={3, 9, 12, 8, 16, 15, 45, 60, 24, 48, 72, 24, 48, 72, 21, 40, 120, 240, 168, 120, 240}.

Mathematica

f = 200; g = 300; h = 30; j = 100; b = {}; Do[a = {}; Do[Do[If[PrimeQ[x^2 + n y^2], AppendTo[a, x^2 + n y^2]], {x, 0, g}], {y, 1, g}]; AppendTo[b, Take[Union[a], h]], {n, 1, f}]; Print[b]; c = {}; Do[a = {}; Do[Do[If[PrimeQ[n^2 + w*n*m + m^2], AppendTo[a, n^2 + w*n*m + m^2]], {n, m, g}], {m, 1, g}]; AppendTo[c, Take[Union[a], h]], {w, 1, j}]; Print[c]; bb = {}; cc = {}; Do[Do[If[b[[p]] == c[[q]], AppendTo[bb, p]; AppendTo[cc, q]], {p, 1, f}], {q, 1, j}]; Union[cc] (*Artur Jasinski*)

Crossrefs

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

Sequence in context: A203168 A284820 A284623 * A030375 A081712 A178663

Adjacent sequences: A139487 A139488 A139489 * A139491 A139492 A139493

Keyword

nonn

Author

Artur Jasinski, Apr 24 2008, Apr 26 2008, Apr 27 2008

Extensions

Edited by N. J. A. Sloane, Apr 25 2008

Extended by T. D. Noe, Apr 27 2009

Typo fixed by Charles R Greathouse IV, Oct 28 2009

Status

approved