A351375 - OEIS

%I #21 Feb 21 2022 21:44:20

%S 0,0,0,174,3,3,3,53,53,25,3,20,93,932,1132,405,2,2,7,6,6,972,39712,2,

%T 2,2,6,6,6,3,1,1,0,0,271,43,3,3,14,14,42,583,131,181,1131,15,93,176,2,

%U 48,685,6,16,16,5,2,2,559,6,6,1351,421,1,1,0,60,12,12,300,29,14,20,53,52

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

%C Conjecture: a(n) exists for any nonnegative integer n.

%C See also Conjecture 1 in A351341.

%H Chai Wah Wu, <a href="/A351375/b351375.txt">Table of n, a(n) for n = 0..10000</a>

%e a(3) = 174 with 3 = 11^5 + 22^5 - 36^3 - 174^3.

%e a(13) = 932 with 13 = 12^5 + 61^5 - 328^3 - 932^3.

%e a(14) = 1132 with 14 = 40^5 + 67^5 - 125^3 - 1132^3.

%e a(15) = 405 with 15 = 31^5 + 36^5 - 283^3 - 405^3.

%e a(21) = 972 with 21 = 37^5 + 64^5 - 608^3 - 972^3.

%e From _Chai Wah Wu_, Feb 21 2022: (Start)

%e a(22) = 39712 with 22 = 106^5 + 599^5 - 24385^3 - 39712^3.

%e a(154) = 22049 with 154 = 163^5 + 432^5 - 16438^3 - 22049^3.

%e a(246) = 22811 with 246 = 161^5 + 412^5 - 4786^3 - 22811^3.

%e a(446) = 230630 with 446 = 843^5 + 1771^5 - 177372^3 - 230630^3.

%e a(769) = 614692 with 769 = 2320^5 + 2777^5 - 46750^3 - 614692^3.

%e (End)

%t FQ[n_]:=IntegerQ[n^(1/5)];

%t tab={};Do[m=0;Label[bb]; k=m^3;Do[If[FQ[n+k+x^3-y^5], tab=Append[tab,m];Goto[aa]],{x,0,m},{y,0,((n+k+x^3)/2)^(1/5)}];m=m+1;Goto[bb];Label[aa],{n,0,21}];Print[tab]

%Y Cf. A000578, A000584, A004842, A004999, A351338, A351341, A351376.

%K nonn

%O 0,4

%A _Zhi-Wei Sun_, Feb 09 2022

%E a(22)-a(73) from _Chai Wah Wu_, Feb 21 2022