A321498 Numbers which can be written in at least two ways in the form (x-y)*(x^2-y^2) with x > y > 0.

45, 63, 81, 96, 99, 117, 128, 135, 153, 160, 171, 175, 189, 192, 207, 224, 225, 243, 256, 261, 275, 279, 288, 297, 315, 320, 325, 333, 351, 352, 360, 369, 375, 384, 387, 405, 416, 423, 425, 432, 441, 448, 459, 475, 477, 480, 495, 504, 512, 513, 525, 531, 539, 544, 549, 567, 575, 576, 585

Offset: 1

Author

Geoffrey B. Campbell and M. F. Hasler, Nov 22 2018

Keywords

• nonn

Comments

An equivalent form is (x - y)^2*(x + y), or d^2*(d + 2y), where d = x - y > 0 and y > 0. See also A321499.

Examples

			   45 = (4 - 1)*(4^2 - 1^2) = (23 - 22)*(23^2 - 22^2),
   63 = (5 - 2)*(5^2 - 2^2) = (32 - 31)*(32^2 - 31^2),
   81 = (6 - 3)*(6^2 - 3^2) = (41 - 40)*(41^2 - 40^2),
   96 = (5 - 1)*(5^2 - 1^2) = (13 - 11)*(13^2 - 11^2),
   99 = (7 - 4)*(7^2 - 4^2) = (50 - 49)*(50^2 - 49^2),
  117 = (8 - 5)*(8^2 - 5^2) = (59 - 58)*(59^2 - 58^2).
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..13937 (terms < 10^5)
• Geoffrey B. Campbell, (m-n)(m^2-n^2) in two different ways, LinkedIn Number Theory Group, Aug. 2018.

Crossrefs

Cf. A321499.

Programs

Mathematica

aQ[n_] := Length[Solve[(x-y)*(x^2-y^2) ==n && x > y && y > 0, {x,y}, Integers]] > 1; Select[Range[600], aQ] (* Amiram Eldar, Dec 06 2018 *)

PARI

select( is_A321498(n,c=2)={n&&!issquarefree(n)&&fordiv(n, d, d^2*(d+2)>n && break; n%d^2&&next; bittest(n\d^2-d, 0)||c--||return(1))}, [0..999]) \\ Define the function is_A321498(). \\ ~30% speed up by David A. Corneth, Nov 23 2018

PARI

is(n) = {if(issquarefree(n), return(0)); if(n % 2 == 0, if(n % 8 == 0, n\=8, return(0))); f = factor(n); e = select(x -> x > 1, f[, 2], 1); if(#e == 0 || n == 1, return(0), k = e[1]); n > f[k, 1]^3} \\ David A. Corneth, Dec 01 2018