## S001059

Number of numbers between (n-1)^5 and n^5 that are not the sum of 6 nonnegative fifth powers.

0, 25, 190, 730, 1998, 4451, 8684, 15402, 25347, 39556, 59021, 84820, 118272, 160901, 214002, 279314, 358366, 452917, 565529, 697455, 850917, 1029699, 1233758, 1466459, 1731541, 2030340, 2367116, 2740580, 3160823, 3626610, 4141371, 4705010, 5327163

1

It is not known what the last term of this sequence is. The first 1000 of these numbers are in S001060. The Mathematica program below computes only the first few terms of this sequence. The computation of these terms required a computer with 64GB of RAM.

T. D. Noe, Plot of 334 terms

T. D. Noe, Table of 334 terms

Wikipedia, Waring’s problem

(Mma) Table[lim = nn^5; t6 = Table[0, {lim}]; Do[num = i^5 + j^5 + k^5 + l^5 + m^5 + n^5; If[0 < num <= lim, t6[[num]]++], {i, 0, nn}, {j, i, nn}, {k, j, nn}, {l, k, nn}, {m, l, nn}, {n, m, nn}]; Length[Select[Flatten[Position[t6, 0]], # > (nn - 1)^5 &]], {nn, 10}]

Cf. S001033-S001060.

nonn,hard

T. D. Noe, Jul 27 2017