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%I #18 Dec 12 2021 12:12:50
%S 3,12,15,17,27,30,40,42,48,60,63,68,75,77,90,95,99,105,108,112,120,
%T 130,135,140,147,153,156,160,165,168,192,195,220,240,243,252,270,272,
%U 273,300,301,308,312,315,323,350,360,363,375,378,380,396,399,420,425,432,448,462,480,495,507
%N Positive numbers x for which x^2 can be expressed as z^2 - y^4 with y != 0.
%C This sequence is closely related to A271576.
%C Conditions to be satisfied for a solution:
%C - z cannot be a square.
%C - z must have at least one prime factor of the form p == 1 (mod 4), a Pythagorean prime (A002144).
%C - If z has prime factors of the form p == 3 (mod 4), which are in A002145, then they must appear in the prime divisor sets of x and y too.
%C - If z is even, x and y must be even too.
%C - The lower limit of the ratio x/y is sqrt(2).
%C Multiple solutions are possible; e.g., term 420 has 5 solutions.
%e The 5 solutions corresponding to a(54) = 420 are 420^2 = 176400 = 444^2 - 12^4 = 580^2 - 20^4 = 609^2 - 21^4 = 1295^2 - 35^4 = 3164^2 - 56^4.
%Y Cf. A000290, A000583, A271576, A002144, A002145.
%K nonn
%O 1,1
%A _Karl-Heinz Hofmann_ and _Hugo Pfoertner_, Dec 09 2021
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