A057102 a(n) = 4 * A073120(n).

24, 96, 120, 240, 336, 384, 480, 720, 840, 960, 1320, 1344, 1536, 1920, 1944, 2016, 2184, 2520, 2880, 3360, 3696, 3840, 3960, 4896, 5280, 5376, 5544, 6144, 6240, 6840, 6864, 7680, 7776, 8064, 8736, 9240, 9360, 9720, 10080, 10296, 10920, 11520, 12144

Offset: 1

Author

Henry Bottomley, Aug 02 2000

Keywords

• nonn
• easy

Comments

This sequence was originally described as the list of "congrua". But that name more properly refers to A256418.

Numbers of the form 4*(x^3*y-x*y^3) (where x,y are integers and x>=y). Squares of these numbers are of the form N^4-K^2 (where N belongs to A135786 and K to A135789 or A135790). Proof uses identity: (4(x^3y-xy^3))^2=(x^2+y^2)^4-(x^4 - 6x^2 y^2 + y^4)^2. - Artur Jasinski, Nov 29 2007, Nov 14 2008

Crossrefs

Cf. A073120, A256418.

Programs

Maple

N:= 10^5: # to get all terms <= N
select(`<=`,{seq(seq(4*(x^3*y-x*y^3),y=1..x-1),x=1..floor(sqrt(N/4+1)))},N);
# If using Maple 11 or earlier, uncomment the following line
# sort(convert(%, list)); # Robert Israel, Apr 06 2015

Mathematica

a = {}; Do[Do[w = 4x^3y - 4x y^3; If[w > 0 && w < 10000, AppendTo[a, w]], {x, y, 1000}], {y, 1, 1000}]; Union[a] (* Artur Jasinski, Nov 29 2007 *)

Extensions

Edited by N. J. A. Sloane, Apr 06 2015 at the suggestion of Robert Israel, Apr 03 2015