A351375 -id:A351375 - cp's OEIS frontend

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.

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, 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, 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

Offset: 0

Zhi-Wei Sun, Feb 09 2022

nonn

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

See also Conjecture 1 in A351341.

			a(4) = 76 with 4 = 775^3 + 1397^3 - (58^5 + 76^5).
a(18) = 129 with 18 = 1693^3 + 3137^3 - (3^5 + 129^5).
a(24) = 136 with 24 = 2534^3 + 3116^3 - (0^5 + 136^5).
a(87) = 81 with 87 = 140^3 + 1658^3 - (64^5 + 81^5).
From _Chai Wah Wu_, Feb 21 2022 : (Start)
a(389) = 3883 with 389 = 590621^3 + 877987^3 - (612^5 + 3883^5).
a(4173) = 3978 with 4173 = 16112^3 + 1108958^3 - (3259^5 + 3978^5).
(End)

Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms 0..100 from Zhi-Wei Sun)

Cf. A000578, A000584, A004842, A004999, A351338, A351341, A351375.

CQ[n_]:=IntegerQ[n^(1/3)];
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]

A351832 Least nonnegative integer m such that n = x^6 + y^6 - (z^2 + m^2) for some nonnegative integers x,y,z with z <= m.

Original entry on oeis.org

0, 0, 0, 6, 6, 20, 7, 7, 19, 24, 24, 7, 6, 6, 5, 5, 7, 26, 26, 6, 6, 22, 9, 5, 5, 6, 98, 6, 6, 6, 5, 5, 4, 4, 32, 5, 5, 26, 5, 4, 4, 20, 322, 7, 4, 4, 3, 3, 4, 4, 22, 3, 3, 22, 3, 3, 2, 2, 418, 2, 2, 2, 1, 1, 0, 0, 94, 6, 23, 20, 7, 19, 24, 20, 20, 7, 6, 22, 5, 7, 19, 18, 18, 6, 22, 37, 59, 5, 6, 24, 24, 6, 6, 21

Offset: 0

Zhi-Wei Sun, Feb 21 2022

nonn

Conjecture: a(n) exists for each nonnegative integer n.

See also Conjecture 2 in A351341.

			a(170) = 2730 with 170 = 9^6 + 15^6 - (2114^2 + 2730^2).
a(5938) = 16184 with 5938 = 17^6 + 25^6 - (2520^2 + 16184^2).
a(9746) = 7600 with 9746 = 11^6 + 21^6 - (5456^2 + 7600^2).

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Cf. A000290, A001014, A001481, A351321, A351338, A351341, A351375, A351376.

QQ[n_]:=IntegerQ[n^(1/6)];
tab={};Do[m=0; Label[bb]; k=m^2; Do[If[QQ[n+k+x^2-y^6], tab=Append[tab,m];Goto[aa]],  {x, 0, m}, {y, 0, ((n+k+x^2)/2)^(1/6)}];m=m+1; Goto[bb]; Label[aa], {n, 0, 100}];Print[tab]

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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.

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