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A266212 Positive integers x such that x^3 = y^4 + z^2 for some positive integers y and z.

%I A266212 #17 Dec 30 2017 09:42:15
%S A266212 8,13,20,25,40,125,128,193,200,208,225,313,320,328,400,500,605,640,
%T A266212 648,1000,1053,1156,1521,1620,1625,1681,1700,2000,2025,2048,2125,2465,
%U A266212 2493,2873,2920,3025,3088,3185,3200,3240,3328,3400,3600,3656,3748,3816,4225,4625,4913,5000,5008,5120,5248,6400,6728,6760,6793,6845,7225,8000
%N A266212 Positive integers x such that x^3 = y^4 + z^2 for some positive integers y and z.
%C A266212 If x^3 = y^4 + z^2, then (a^(4k)*x)^3 = (a^(3k)*y)^4 + (a^(6k)*z)^2 for all a = 1,2,3,... and k = 0,1,2,... So the sequence has infinitely many terms.
%C A266212 Conjecture: For any integer m, there are infinitely many triples (x,y,z) of positive integers with x^4 - y^3 + z^2 = m.
%C A266212 This is stronger than the conjecture in A266152.
%H A266212 Zhi-Wei Sun and Chai Wah Wu, <a href="/A266212/b266212.txt">Table of n, a(n) for n = 1..698</a> n = 1..100 from Zhi-Wei Sun
%H A266212 Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/179b.pdf">New conjectures on representations of integers (I)</a>, Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
%e A266212 a(1) = 8 since 8^3 = 4^4 + 16^2.
%e A266212 a(2) = 13 since 13^3 = 3^4 + 46^2.
%e A266212 a(3) = 20 since 20^3 = 4^4 + 88^2.
%e A266212 a(8) = 193 since 193^3 = 6^4 + 2681^2.
%e A266212 a(12) = 313 since 313^3 = 66^4 + 3419^2.
%e A266212 a(20) = 1000 since 1000^3 = 100^4 + 30000^2.
%t A266212 SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]]
%t A266212 n=0;Do[Do[If[SQ[x^3-y^4],n=n+1;Print[n," ",x];Goto[aa]],{y,1,x^(3/4)}];Label[aa];Continue,{x,1,8000}]
%Y A266212 Cf. A000290, A000578, A000583, A262827, A266003, A266004, A266152, A266153.
%K A266212 nonn
%O A266212 1,1
%A A266212 _Zhi-Wei Sun_, Dec 23 2015