Numbers that are not the sum of 7 nonnegative fifth powers.

8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 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, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86

1

There are a finite, but unknown number of such numbers.

T. D. Noe, Plot of 1000 terms

T. D. Noe, Table of 1000 terms

Wikipedia, Waring’s problem

(Mma) nn = 5; lim = nn^5; t7 = Table[0, {lim}]; Do[num = i^5 + j^5 + k^5 + l^5 + m^5 + n^5 + o^5; If[0 < num <= lim, t7[[num]]++], {i, 0, nn}, {j, i, nn}, {k, j, nn}, {l, k, nn}, {m, l, nn}, {n, m, nn}, {o, n, nn}]; Take[Flatten[Position[t7, 0]], 1000]

nonn,fini

T. D. Noe, Jul 04 2017