Numbers that are not the sum of 6 nonnegative fifth powers.

7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84

1

There may be a finite, but very large, number of terms.

T. D. Noe, Plot of 1000 terms

T. D. Noe, Table of 1000 terms

Wikipedia, Waring’s problem

(Mma) nn = 5; 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}]; Take[Flatten[Position[t6, 0]], 1000]

nonn

T. D. Noe, Jul 27 2017