## S001051

Numbers that are not the sum of 24 nonnegative 5-th powers.

25, 26, 27, 28, 29, 30, 31, 56, 57, 58, 59, 60, 61, 62, 63, 87, 88, 89, 90, 91, 92, 93, 94, 95, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191

1

There are 190 terms in this sequence, ending with 7138, which means that all numbers greater than 7138 can be written as the sum of 24 nonnegative 5-th powers.

T. D. Noe, Plot of 190 terms

T. D. Noe, Table of 190 terms

Wikipedia, Waring’s problem

(Mma) nn = 6; lim = nn^5; t24 = Table[0, {lim}]; Do[num = i^5 + j^5 + k^5 + l^5 + m^5 + n^5 + o^5 + p^5 + q^5 + r^5 + s^5 + t^5 + u^5 + v^5 + w^5 + x^5 + y^5 + z^5 + a^5 + b^5 + c^5 + d^5 + e^5 + f^5; If[0 < num <= lim, t24[[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}, {q, p, nn}, {r, q, nn}, {s, r, nn}, {t, s, nn}, {u, t, nn}, {v, u, nn}, {w, v, nn}, {x, w, nn}, {y, x, nn}, {z, y, nn}, {a, z, nn}, {b, a, nn}, {c, b, nn}, {d, c, nn}, {e, d, nn}, {f, e, nn}]; Flatten[Position[t24, 0]]

Cf. S001033-S001060.

nonn,fini

T. D. Noe, Jul 04 2017