A088242 Values of x, where x^2 + xy + y^2 = p (x<y) is a prime of the form 6n + 1 (A002476).

1, 1, 2, 1, 3, 1, 4, 2, 1, 3, 3, 2, 5, 6, 3, 5, 1, 3, 4, 7, 2, 1, 6, 5, 1, 9, 7, 6, 1, 3, 10, 8, 3, 9, 4, 7, 11, 8, 1, 11, 5, 7, 1, 2, 7, 9, 4, 13, 5, 8, 1, 3, 9, 5, 14, 11, 9, 8, 11, 3, 13, 12, 7, 10, 1, 15, 2, 6, 14, 13, 4, 10, 3, 13, 7, 17, 3, 7, 9, 13, 8, 11, 16, 15, 6, 3, 12, 17, 7, 9, 1, 3, 16

Offset

1,3

Links

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

Bat-Od Battseren and Bayarmagnai Gombodorj, A quadratic form of p = 3k+1 primes, arXiv:2511.19763 [math.NT], 2025. See Table p. 3.

Maple

R:= NULL: count:= 0:
for k from 1 while count < 100 do
  p:= 6*k+1;
  if not isprime(p) then next fi;
  S:= select(t -> subs(t, x) > 0 and subs(t, x) < subs(t, y), [isolve(x^2+x*y+y^2=p)]);
  S:= map(t -> subs(t, x), S);
   R:= R, op(S); count:= count+1;
od:
R; # Robert Israel, Jun 16 2025

Mathematica

Reap[For[n = 1, n <= 200, n++, If[PrimeQ[p = 6 n + 1], s = Solve[x^2 + x y + y^2 == p && 0 < x < y, {x, y}, Integers]; Sow[x /. s[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 07 2020 *)

Crossrefs

Cf. A002476, A088241, A088243, A088296, A088298, A088299, A088977.

Sequence in context: A340088 A087114 A303753 * A228814 A299483 A113398

Adjacent sequences: A088239 A088240 A088241 * A088243 A088244 A088245

Keyword

nonn

Author

Lekraj Beedassy, Nov 03 2003

Extensions

More terms from Ray Chandler, Nov 04 2003

Status

approved