A259263 Numbers of the form (m*k)^2/(m^2-k^2) for distinct integers m and k.

12, 18, 48, 72, 108, 147, 150, 162, 180, 192, 225, 240, 288, 300, 400, 405, 432, 448, 450, 578, 588, 600, 648, 720, 768, 882, 900, 960, 972, 980, 1008, 1100, 1152, 1200, 1260, 1323, 1350, 1452, 1458, 1600, 1620, 1728, 1792, 1800, 2025, 2028, 2100, 2160, 2178, 2312, 2352, 2400, 2592, 2700, 2880, 3042, 3072, 3150

Offset: 1

Author

Derek Orr, Jun 22 2015

Keywords

• nonn

Comments

The odd numbers are much more rare than even numbers: 147, 225, 405, 1323, 2025, 3645, 3675, ... For 1 <= m <= 10^4 and 1 <= k <= m, there are 9217 total solutions. Of these solutions, only 679 are odd. See A259288.

Similarly, the reciprocals of these numbers can be represented as the difference in the reciprocals of two squares (i.e., there exists two distinct integers m and k satisfying 1/a(n) = 1/m^2 - 1/k^2).

If a(n) is a square, its square root is in A111200.

Examples

			(3*6)^2/(6^2-3^2) = 18^2/(3*9) = 12. So 12 is a member of this sequence.
		

Crossrefs

Cf. A063664, A111200, A259288.

Programs

PARI

v=[];for(m=1,7500,for(n=1,m-1,if(type(s=(m*n)^2/(m^2-n^2))=="t_INT",v=concat(v,s))));vecsort(v,,8)