## S001052

Numbers that are not the sum of 25 nonnegative 5-th powers.

26, 27, 28, 29, 30, 31, 57, 58, 59, 60, 61, 62, 63, 88, 89, 90, 91, 92, 93, 94, 95, 119, 120, 121, 122, 123, 124, 125, 126, 127, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 212, 213, 214, 215, 216

1

There are 126 terms in this sequence, ending with 6123, which means that all numbers greater than 6123 can be written as the sum of 25 nonnegative 5-th powers.

T. D. Noe, Plot of 126 terms

T. D. Noe, Table of 126 terms

Wikipedia, Waring’s problem

(Mma) nn = 6; lim = nn^5; t25 = 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 + g^5; If[0 < num <= lim, t25[[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}, {g, f, nn}]; Flatten[Position[t25, 0]]

Cf. S001033-S001060.

nonn,fini

T. D. Noe, Jul 04 2017

© Tony D Noe 2014-2017