Numbers not represented by the sum of 7 positive cubes.

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 34, 36, 37, 38, 39, 41, 43, 44, 45, 46, 48, 50, 51, 52, 53, 55, 57, 58, 60, 62, 63, 64, 65, 67, 69, 71, 72, 74, 76, 78, 79, 81, 82, 83, 86, 88, 89, 90, 93, 95

1

For positive integers, rather than nonnegative, there are always integers that cannot be represented no matter how many terms are included.

T. D. Noe, Plot of 208 terms

T. D. Noe, Table of 208 terms

Wikipedia, Waring’s problem

(Mma) nn = 20; t = Select[Union[Flatten[Table[i^3 + j^3 + k^3 + l^3 + m^3 + n^3 + o^3, {i, nn}, {j, i, nn}, {k, j, nn}, {l, k, nn}, {m, l, nn}, {n, m, nn}, {o, n, nn}]]], # <= nn^3 + 6 &]; Complement[Range[t[[-1]]], t]

Cf. A018890.

nonn,full

T. D. Noe, May 18 2017