Number of numbers between n-1 and n that are the sum of consecutive sixth powers.

0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 1, 9, 10, 10, 5, 9, 13, 13, 7, 11, 15, 17, 5, 15, 18, 19, 6, 19, 21, 7, 19, 22, 11, 20, 24, 16, 16, 25, 28, 8, 27, 27, 8, 29, 32, 8, 31, 14, 27, 33, 12, 30, 36, 12, 33, 35, 12, 37, 39, 9, 37, 17, 36, 41, 14, 38, 44, 11, 43, 17, 40

1

At higher powers, it is assumed that the plot becomes even more complex.

T. D. Noe, Plot of 10000 terms

T. D. Noe, Table of 10000 terms

(Mma) nMax = 100; pwr = 6; t = Reap[Do[k = n; s = 0; While[s = s + k^pwr; s <= nMax^pwr, Sow[s]; k++], {n, nMax}]]; t = Union[t[[2, 1]]]; t = Complement[t, Range[0, nMax]^pwr]; ta = Tally[Floor[t^(1/pwr)]]; tb = Complement[Range[0, nMax - 1], Transpose[ta][[1]]]; tc = Transpose[{tb, Table[0, {Length[tb]}]}]; Transpose[Union[ta, tc]][[2]]

nonn

T. D. Noe, May 14 2015