cp's OEIS Frontend

A351341 Least nonnegative integer m such that n = x^4 + y^4 - (z^3 + m^3) for some nonnegative integers x,y,z with z <= m.

%I A351341 #24 Feb 20 2022 23:06:48
%S A351341 0,0,0,63,3,3,4,2,2,2,4,21,37,6,1,1,0,0,4,11,7,14,5,2,2,4,8,3,3,5,1,1,
%T A351341 0,4,4,45,5,5,11,6,6,6,32,3,7,11,3,3,6,8,8,48,13,3,3,3,6,6,31,20,93,
%U A351341 55,3,49,33,2,2,5,5,3,3,4,2,2,2,69,17,29,11,1,1,0,0,5,61,29,8,5,2,2,4,21,29,51,6,1,1,0,4,85,13
%N A351341 Least nonnegative integer m such that n = x^4 + y^4 - (z^3 + m^3) for some nonnegative integers x,y,z with z <= m.
%C A351341 Conjecture 1: Let k be 4 or 5. Then each integer can be written as x^k + y^k - (z^3 + w^3) with x,y,z,w nonnegative integers.
%C A351341 Two examples for k = 5: -4 = 58^5 + 76^5 - (775^3 + 1397^3) and 14 = 40^5 + 67^5 - (125^3 + 1132^3).
%C A351341 Conjecture 2: Let k be among 4, 5, 6 and 7. Then any integer can be written as x^k + y^k - (z^2 + w^2) with x,y,z,w nonnegative integers.
%C A351341 Examples for k = 6, 7: 170 = 9^6 + 15^6 - (2114^2 + 2730^2) and 469 = 7^7 + 8^7 - (1001^2 + 1385^2).
%C A351341 Conjecture 3: For any integer k > 3, there are no nonnegative integers x,y,z,w such that x^k + y^k - (z^k + w^k) = 3.
%C A351341 See also another similar conjecture in A351338.
%H A351341 Zhi-Wei Sun, <a href="/A351341/b351341.txt">Table of n, a(n) for n = 0..10000</a>
%e A351341 a(60) = 93 with 60 = 25^4 + 27^4 - (49^3 + 93^3).
%e A351341 a(527) = 527 with 527 = 29^4 + 110^4 - (91^3 + 527^3).
%e A351341 a(2198) = 1704 with 2198 = 85^4 + 304^4 - (1539^3 + 1704^3).
%e A351341 a(4843) = 1965 with 4843 = 142^4 + 338^4 - (1804^3 + 1965^3).
%t A351341 QQ[n_]:=IntegerQ[n^(1/4)];
%t A351341 tab={};Do[m=0;Label[bb]; k=m^3;Do[If[QQ[n+k+x^3-y^4], tab=Append[tab,m];Goto[aa]],{x,0,m},{y,0,((n+k+x^3)/2)^(1/4)}];m=m+1;Goto[bb];Label[aa],{n, 0, 100}];Print[tab]
%Y A351341 Cf. A000578, A000583, A000584, A001014, A001015, A001481, A004831, A004999, A351306, A351321, A351338.
%K A351341 nonn
%O A351341 0,4
%A A351341 _Zhi-Wei Sun_, Feb 08 2022