A061408 For each y >= 1 there are only finitely many values of x >= 1 such that x-y and x+y are both positive squares; list all such pairs (x,y) ordered by values of y; sequence gives y values.

4, 6, 8, 10, 12, 12, 14, 16, 16, 18, 20, 20, 22, 24, 24, 24, 26, 28, 28, 30, 30, 32, 32, 34, 36, 36, 36, 38, 40, 40, 40, 42, 42, 44, 44, 46, 48, 48, 48, 48, 50, 52, 52, 54, 54, 56, 56, 56, 58, 60, 60, 60, 60, 62, 64, 64, 64, 66, 66, 68, 68, 70, 70, 72

Offset: 0

Author

Jason Earls, May 01 2001

Keywords

• nonn

Comments

Each even integer y >= 4 occurs A056924(y/2) times. - Robert Israel, Dec 10 2017

Examples

			Pairs are [5, 4], [10, 6], [17, 8], [26, 10], [13, 12], [37, 12], [50, 14], ... For example, 5-4 = 1^2, 5+4 = 3^2.
		

References

• Donald D. Spencer, Computers in Number Theory, Computer Science Press, Rockville MD, 1982, pp. 130-131.

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A056924, A061409, A060829, A060830.

Programs

Maple

seq(y $ nops(select(t -> (t^2 < y/2), numtheory:-divisors(y/2))), y=2..100,2); # Robert Israel, Dec 10 2017

Mathematica

Table[Table[y, {Select[Divisors[y/2], #^2 < y/2&] // Length}], {y, 2, 100, 2}] // Flatten (* Jean-François Alcover, Feb 28 2019, after Robert Israel *)

Formula

The solutions are given by x = r^2 + 2*r*k + 2*k^2, y = 2*k*(k+r) with r >= 1, k >= 1. - N. J. A. Sloane, May 02 2001

Extensions

Definition clarified by Robert Israel, Dec 10 2017