login
Numbers of the form x^5 + 10*x^3*y^2 + 5*x*y^4 (where x,y are integers).
0

%I #12 Sep 25 2024 23:12:43

%S 16,121,122,496,512,528,1441,1562,1563,1684,3376,3872,3888,3904,4400,

%T 6841,8282,8403,8404,8525,9966,12496,15872,16368,16384,16400,16896,

%U 20272,21121,27962,29403,29524,29525,29646,31087,33616,37928,46112

%N Numbers of the form x^5 + 10*x^3*y^2 + 5*x*y^4 (where x,y are integers).

%C Squares of these numbers are of the form N^5+M^2 (where N belongs to A000404 and M to A135795). Proof uses: (x^5+10x^3 y^2+5xy^4)^2=(x^2-y^2)^5+(5x^4y+10x^2y^3+y^5)^2.

%C Also numbers of the form ((y + x)^5 - (y - x)^5)/2 = x^5 + 10*x^3*y^2 + 5*x*y^4. - _Artur Jasinski_, Oct 10 2008

%C Refers to A057102, which had an incorrect description and has been replaced by A256418.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^5 + 10x^3 y^2 + 5x y^4; If[w < 100000, AppendTo[a, w]], {x, 1, 1000}], {y, 1, 1000}]; Union[a]

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

%K nonn

%O 1,1

%A _Artur Jasinski_, Nov 29 2007, Oct 10 2008