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A351376 Least nonnegative integer m such that n = x^3 + y^3 - (z^5 + m^5) for some nonnegative integers x,y,z with z <= m.

%I A351376 #14 Aug 04 2022 05:21:06
%S A351376 0,0,0,2,76,3,1,1,0,0,6,5,4,7,1,1,0,51,129,14,22,2,2,4,136,1,1,0,0,27,
%T A351376 7,2,2,1,1,0,3,3,14,2,2,44,11,5,8,6,101,4,4,28,14,6,1,1,0,17,42,33,2,
%U A351376 2,20,2,1,1,0,0,3,8,3,2,1,1,0,3,6,41,3,43,12,10,10,6,6,6,59,29,33,81,2,1,1,0,2,2,2,2,2,3,3,3,2
%N A351376 Least nonnegative integer m such that n = x^3 + y^3 - (z^5 + m^5) for some nonnegative integers x,y,z with z <= m.
%C A351376 Conjecture: a(n) exists for any nonnegative integer n.
%C A351376 See also Conjecture 1 in A351341.
%H A351376 Chai Wah Wu, <a href="/A351376/b351376.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..100 from Zhi-Wei Sun)
%e A351376 a(4) = 76 with 4 = 775^3 + 1397^3 - (58^5 + 76^5).
%e A351376 a(18) = 129 with 18 = 1693^3 + 3137^3 - (3^5 + 129^5).
%e A351376 a(24) = 136 with 24 = 2534^3 + 3116^3 - (0^5 + 136^5).
%e A351376 a(87) = 81 with 87 = 140^3 + 1658^3 - (64^5 + 81^5).
%e A351376 From _Chai Wah Wu_, Feb 21 2022 : (Start)
%e A351376 a(389) = 3883 with 389 = 590621^3 + 877987^3 - (612^5 + 3883^5).
%e A351376 a(4173) = 3978 with 4173 = 16112^3 + 1108958^3 - (3259^5 + 3978^5).
%e A351376 (End)
%t A351376 CQ[n_]:=IntegerQ[n^(1/3)];
%t A351376 tab={};Do[m=0;Label[bb]; k=m^5;Do[If[CQ[n+k+x^5-y^3], tab=Append[tab,m];Goto[aa]],{x,0,m},{y,0,((n+k+x^5)/2)^(1/3)}];m=m+1;Goto[bb];Label[aa],{n,0,100}];Print[tab]
%Y A351376 Cf. A000578, A000584, A004842, A004999, A351338, A351341, A351375.
%K A351376 nonn
%O A351376 0,4
%A A351376 _Zhi-Wei Sun_, Feb 09 2022