A351375 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.

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

Offset

0,4

Comments

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

See also Conjecture 1 in A351341.

Links

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Example

a(3) = 174 with 3 = 11^5 + 22^5 - 36^3 - 174^3.
a(13) = 932 with 13 = 12^5 + 61^5 - 328^3 - 932^3.
a(14) = 1132 with 14 = 40^5 + 67^5 - 125^3 - 1132^3.
a(15) = 405 with 15 = 31^5 + 36^5 - 283^3 - 405^3.
a(21) = 972 with 21 = 37^5 + 64^5 - 608^3 - 972^3.
From Chai Wah Wu, Feb 21 2022: (Start)
a(22) = 39712 with 22 = 106^5 + 599^5 - 24385^3 - 39712^3.
a(154) = 22049 with 154 = 163^5 + 432^5 - 16438^3 - 22049^3.
a(246) = 22811 with 246 = 161^5 + 412^5 - 4786^3 - 22811^3.
a(446) = 230630 with 446 = 843^5 + 1771^5 - 177372^3 - 230630^3.
a(769) = 614692 with 769 = 2320^5 + 2777^5 - 46750^3 - 614692^3.
(End)

Mathematica

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

Crossrefs

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

Sequence in context: A138267 A096709 A180417 * A160660 A025390 A377875

Adjacent sequences: A351372 A351373 A351374 * A351376 A351377 A351378

Keyword

nonn

Author

Zhi-Wei Sun, Feb 09 2022

Extensions

a(22)-a(73) from Chai Wah Wu, Feb 21 2022

Status

approved