S001055

Number of numbers between (n-1)^5 and n^5 that are not the sum of 7 nonnegative fifth powers.

0, 24, 183, 710, 1941, 4320, 8402, 14869, 24366, 37951, 56352, 80556, 111817, 151403, 200233, 259959, 331841, 416457, 517262, 633399, 767766, 924596, 1098895, 1295231, 1520154, 1768478, 2046574, 2347305, 2690085, 3066670, 3469706, 3903408, 4383843

1

It is not known what the last term of this sequence is. However, an upper bound is probably 10^15. The first 1000 of these numbers are in S001056. The Mathematica program below computes only the first few terms of this sequence.

T. D. Noe, Plot of 280 terms

T. D. Noe, Table of 280 terms

Wikipedia, Waring’s problem

(Mma) Table[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}]; Length[Select[Flatten[Position[t7, 0]], # > (nn - 1)^5 &]], {nn, 10}]

Cf. S001033.

nonn,hard,fini

T. D. Noe, Jul 04 2017