login
Numbers of the form x^4 + 6x^2 y^2 + y^4 (where x,y are positive integers).
1

%I #7 Apr 06 2015 10:47:28

%S 8,41,128,136,313,353,648,656,776,1201,1241,1513,2048,2056,2176,2696,

%T 3281,3321,3593,4481,5000,5008,5128,5648,7048,7321,7361,7633,8521

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

%C 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+6x^2y^2+y^4)^2=(x^2-y^2)^4+(4x^3y+4x*y^3)^2(*Artur Jasinski*)

%C Refers to A057102, which had an incorrect description and has been replaced by A256418.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

%t 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] (*Artur Jasinski*)

%t Union[Select[#[[1]]^4+6#[[1]]^2 #[[2]]^2+#[[2]]^4&/@Tuples[Range[ 1000],2], #<10000&]] (* _Harvey P. Dale_, Oct 07 2012 *)

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

%K nonn

%O 1,1

%A _Artur Jasinski_, Nov 29 2007