Irregular triangle of numbers requiring 37-n positive fifth powers to represent.

223, 191, 222, 159, 190, 221, 127, 158, 189, 220, 95, 126, 157, 188, 219, 63, 94, 125, 156, 187, 218, 466, 31, 62, 93, 124, 155, 186, 217, 434, 465, 30, 61, 92, 123, 154, 185, 216, 402, 433, 464, 29, 60, 91, 122, 153, 184, 215, 370, 401, 432, 463, 28, 59, 90, 121

0

The lengths of the row are 1, 2, 3, 4, 5, 7, 9, 10, 11, 14, 22, 38, 64, and 107. The list was stopped when terms exceeded 10000. The sequence A188462, the least number of 5th powers to represent k, has 10000 terms which were used to construct this sequence.

T. D. Noe, Plot of 14 rows of terms

T. D. Noe, Table of 14 rows of terms

Wikipedia, Waring’s problem

(Mma) Cnt5[n_] := Module[{k = 1}, While[Length[PowersRepresentations[n, k, 5]] == 0, k++]; k]; Monitor[t3 = {}; Do[AppendTo[t3, Cnt5[n]], {n, 10000}], n]; Table[Flatten[Position[t3, i]], {i, 37, 24, -1}]

nonn,hard,fini

T. D. Noe, Jul 04 2017