Number of numbers between (n-1)^5 and n^5 that are not the sum of 8 nonnegative fifth powers.

0, 23, 176, 686, 1869, 4146, 7986, 14032, 22747, 35179, 51482, 72413, 98996, 131675, 170439, 216269, 270986, 330572, 400541, 475144, 559920, 656353, 751747, 850045, 965800, 1078100, 1196913, 1310691, 1446152, 1589502, 1704328, 1816850, 1947816, 2041543

1

There are a total of 108635543 numbers that are not the sum of 8 nonnegative fifth powers. The last such number is 68578904422. The first 1000 of these numbers are in S001043; the last 1000 are in S001044. The Mathematica program below computes only the first few terms of this sequence. The Mathematica program is useful for computing only a few terms.

T. D. Noe, Plot of 147 terms

T. D. Noe, Table of 200 terms

Wikipedia, Waring’s problem

(Mma) Table[lim = nn^5; t8 = Table[0, {lim}]; Do[num = i^5 + j^5 + k^5 + l^5 + m^5 + n^5 + o^5 + p^5; If[0 < num <= lim, t8[[num]]++], {i, 0, nn}, {j, i, nn}, {k, j, nn}, {l, k, nn}, {m, l, nn}, {n, m, nn}, {o, n, nn}, {p, o, nn}]; Length[Select[Flatten[Position[t8, 0]], # > (nn - 1)^5 &]], {nn, 10}]

nonn,fini,hard

T. D. Noe, Jun 14 2017