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%I #10 Dec 22 2015 10:18:09
%S 3,3,2,6,13,2,3,5,5,3,28,4,15,4,10,33,3,7,5,238,31,3,4,5,3,11,4,5,21,
%T 11,6,4,17,11,5,98,7,4,4,5,147,19,5,4,5,6,4,29,75,1011,7,9,7,4,8,6,59,
%U 47,4,5,71,4,17,45,13,7,18,9,175,8
%N Least positive integer y such that -n = x^4 - y^3 + z^2 for some positive integers x and z, or 0 if no such y exists.
%C The conjecture in A266152 implies that a(n) > 0 for all n > 0.
%C It seems that a(n) < n*(n+4)/2 for all n > 1.
%H Zhi-Wei Sun, <a href="/A266153/b266153.txt">Table of n, a(n) for n = 1..10000</a>
%e a(1) = 3 since -1 = 1^4 - 3^3 + 5^2.
%e a(2) = 3 since -2 = 2^4 - 3^3 + 3^2.
%e a(11) = 28 since -11 = 5^4 - 28^3 + 146^2.
%e a(20) = 238 since -20 = 32^4 - 238^3 + 3526^2.
%t SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]]
%t Do[y=Floor[n^(1/3)]+1;Label[bb];Do[If[SQ[-n+y^3-x^4],Print[n," ",y];Goto[aa]],{x,1,(-n+y^3)^(1/4)}];y=y+1;Goto[bb];Label[aa];Continue,{n,1,70}]
%Y Cf. A000290, A000578, A000583, A262827, A266004, A266152.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Dec 22 2015
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