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%I #17 Dec 25 2015 20:54:15
%S 3,1,3703,5,43,2,119,3,1,19,5,384,2,29,29,1,7,18,6,3,13,14,869,7,2,15,
%T 3,1,10,5,23,2,20,10,1,45,6,2373,4,1193,5,52,7,36,54,3,18,5,13,4,2,
%U 385,9,1,14,6,3,76,250,250,34,2,8,3,1,336,5,52,2,8,28,1,21,12,13,4,113
%N Least positive integer x such that n + x^2 = y^3 + z^3 for some positive integers y and z, or 0 if no such x exists.
%C Conjecture: For any integer m, there are positive integers x, y and z such that m + x^2 = y^3 + z^3.
%C This is similar to the conjecture in A266152. We have verified it for all integers m with |m| <= 25000.
%C Obviously, a(k^3) = 1 for any positive integer k.
%C See also A266231 for a related sequence.
%H Zhi-Wei Sun, <a href="/A266230/b266230.txt">Table of n, a(n) for n = 0..10000</a>
%H Zhi-Wei Sun, <a href="/A266230/a266230_1.txt">Checking the conjecture for integers m with 10000 < |m| <= 25000</a>
%e a(0) = 3 since 0 + 3^2 = 1^3 + 2^3.
%e a(2) = 3703 since 2 + 3703^2 = 107^3 + 232^3.
%e a(3) = 5 since 3 + 5^2 = 1^3 + 3^3.
%e a(4) = 43 since 4 + 43^2 = 5^3 + 12^3.
%e a(37) = 2373 since 37 + 2373^2 = 93^3 + 169^3.
%e a(1227) = 132316 since 1227 + 132316^2 = 1874^3 + 2219^3.
%t CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]
%t Do[x=1;Label[bb];Do[If[CQ[n+x^2-y^3],Print[n," ",x];Goto[aa]],{y,1,((n+x^2)/2)^(1/3)}];x=x+1;Goto[bb];Label[aa];Continue,{n,0,80}]
%Y Cf. A000290, A000578, A003325, A266152, A266153, A266212, A266231.
%K nonn
%O 0,1
%A _Zhi-Wei Sun_, Dec 24 2015
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