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A061409 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 x values.

%I A061409 #18 Dec 15 2017 17:35:02
%S A061409 5,10,17,26,13,37,50,20,65,82,29,101,122,25,40,145,170,53,197,34,226,
%T A061409 68,257,290,45,85,325,362,41,104,401,58,442,125,485,530,52,73,148,577,
%U A061409 626,173,677,90,730,65,200,785,842,61,109,229,901,962
%N A061409 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 x values.
%D A061409 Donald D. Spencer, Computers in Number Theory, Computer Science Press, Rockville MD, 1982, pp. 130-131.
%H A061409 Robert Israel, <a href="/A061409/b061409.txt">Table of n, a(n) for n = 0..10000</a>
%F A061409 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
%e A061409 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.
%p A061409 seq(op(sort(map(k -> (k^2 + (y/2/k)^2), select(t -> t^2 < y/2, convert(numtheory:-divisors(y/2),list))))),y=2..100,2); # _Robert Israel_, Dec 10 2017
%Y A061409 Cf. A061408, A060829, A060830.
%K A061409 nonn
%O A061409 0,1
%A A061409 _Jason Earls_, May 01 2001
%E A061409 Definition clarified by _Robert Israel_, Dec 10 2017