A228880 Numbers of the form x^2*y*(2*x + y).

0, 3, 8, 15, 20, 24, 35, 48, 63, 80, 84, 99, 120, 128, 143, 144, 168, 180, 195, 224, 240, 243, 255, 275, 288, 308, 320, 323, 360, 384, 399, 440, 468, 483, 495, 528, 560, 575, 600, 624, 648, 660, 675, 728, 735, 768, 783, 819, 840, 884, 899, 960, 975, 1008

Offset: 1

Author

Arkadiusz Wesolowski, Sep 11 2013

Keywords

• nonn

Comments

(y^2 + 2*x*y - x^2)^4 + (2*x + y)*x^2*y*(2*x + 2*y)^4 = (x^4 + y^4 + 10*x^2*y^2 + 4*x*y^3 + 13*x^3*y)^2. The equation implies that for any n, x^4 + a(n)*y^4 = z^2 is solvable in integers.

References

• L. E. Dickson, History of the Theory of Numbers, Vol. II. Diophantine analysis, Carnegie Institute of Washington, 1919. Reprinted by AMS Chelsea Publishing, New York, 1992, p. 631.

Crossrefs

Cf. A218381.

Programs

Mathematica

n = 1008; limx = Floor[(n/2)^(1/3)]; limy = Floor@Sqrt[n]; Select[Union@Flatten@Table[x^2*y*(2*x + y), {x, 0, limx}, {y, limy}], # <= n &]