A135797 Numbers of the form x^4 + 6*x^2*y^2 + y^4 (where x,y are positive integers).

8, 41, 128, 136, 313, 353, 648, 656, 776, 1201, 1241, 1513, 2048, 2056, 2176, 2696, 3281, 3321, 3593, 4481, 5000, 5008, 5128, 5648, 7048, 7321, 7361, 7633, 8521, 10368, 10376, 10496, 10601, 11016, 12416, 14281, 14321, 14593, 15368, 15481, 17561, 19208, 19216, 19336, 19856, 21256, 21601, 24208

Offset

1,1

Comments

Squares of these numbers are of the form N^4-M^2 (where N belongs to A135786 and M to A135796). Proof uses: (x^4+6*x^2*y^2+y^4)^2 = (x^2-y^2)^4+(4*x^3*y+4*x*y^3)^2.

Refers to A057102, which had an incorrect description and has been replaced by A256418. As a result the present sequence should be re-checked. - N. J. A. Sloane, Apr 06 2015

Links

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

Maple

N:= 40000: # for terms <= N
S:= {}:
for x from 1 while x^4 + 6*x^2 + 1 <= N do
  for y from 1 do
    v:= x^4 + 6*x^2*y^2 + y^4;
    if v > N then break fi;
    S:= S union {v}
od od:
sort(convert(S, list)); # Robert Israel, Mar 02 2026

Mathematica

a = {}; Do[Do[w = x^4 + 6x^2 y^2 + y^4; If[w < 10000, AppendTo[a, w]], {x, y, 1000}], {y, 1, 1000}]; Union[a]
(* alternate program *)
Union[Select[#[[1]]^4+6#[[1]]^2 #[[2]]^2+#[[2]]^4&/@Tuples[Range[ 1000], 2], #<10000&]] (* Harvey P. Dale, Oct 07 2012 *)

Crossrefs

Cf. A000404, A050803, A057102, A135784, A060803, A135786, A135787, A135789, A135790, A135791, A135792, A135793, A135795, A135796, A135796.

Sequence in context: A326286 A250322 A383854 * A171714 A342034 A304160

Adjacent sequences: A135794 A135795 A135796 * A135798 A135799 A135800

Keyword

nonn,more

Author

Artur Jasinski, Nov 29 2007

Extensions

More terms from Robert Israel, Mar 02 2026

Status

approved