Number beginning n consecutive numbers that are not free of fourth powers.

16, 80, 33614, 202099373

1

No other terms less than 10^9.

T. D. Noe, Plot of 4 terms

Example: 80 = 2^4*5 and 81 = 3^4 are the first two consecutive numbers that are not free of fourth powers.

(Mma) s = Select[Range[10^7], Max[Transpose[FactorInteger[#]][[2]]] > 3 &]; d = Differences[s]; t = {16}; n = 0; While[n++; i = Position[Partition[d, n, 1], Table[1, {n}], 1, 1]; i != {}, i = Flatten[i]; AppendTo[t, s[[i[[1]]]]]]; t

Cf. A045882 (n consecutive numbers not squarefree), A046101 (numbers not free of fourth powers), S000163.

nonn,hard,more

T. D. Noe, Jul 23 2014