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%I #13 Dec 30 2017 09:42:30
%S 3,13,27,147,203,5507,15661,16957,21531,29931,38051,47171,57147,84027,
%T 85547,90891,167051,273651,337501,392881,421715,566691,609971,698113,
%U 914701,1229283,1435213,1564573,1786587,1987571,2523387,2579377,2716443,3760347,3778273
%N Positive integers x such that x^3 - 1 = y^4 + z^2 for some positive integers y and z.
%C The conjecture in A266212 implies that this sequence has infinitely many terms.
%H 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 a(1) = 3 since 3^3 - 1 = 1^4 + 5^2.
%e a(2) = 13 since 13^3 - 1 = 6^4 + 30^2.
%e a(6) = 5507 since 5507^3 - 1 = 29^4 + 408669^2.
%e a(16) = 90891 since 90891^3 - 1 = 949^4 + 27387137^2.
%e a(35) = 3778273 since 3778273^3 - 1 = 85386^4 + 883654380^2.
%t SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]]
%t n=0;Do[Do[If[SQ[x^3-1-y^4],n=n+1;Print[n," ",x];Goto[aa]],{y,1,(x^3-1)^(1/4)}];Label[aa];Continue,{x,1,10^5}]
%Y Cf. A000290, A000578, A000583, A262827, A266003, A266004, A266152, A266153, A266212.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Dec 24 2015
%E a(17)-a(35) from _Lars Blomberg_, Dec 30 2015
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