Numbers n such that binomial(2n,n) is fifth-power free.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 60, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

1

The paper by Sander deals only with the square-free case.

T. D. Noe, Plot of 2243 terms

T. D. Noe, Table of 2243 terms

J. W. Sander, A story of binomial coefficients and primes, Amer. Math. Monthly 102 (1995), 802-807.

(Mma) t = Table[f = FactorInteger[Binomial[2*n, n]]; s = Select[f, #[[2]] > 4 &]; If[s == {}, 0, s[[-1, 1]]], {n, 1000}]; Flatten[Position[t, 0]]

Cf. A110495 (cube-free), S001082 (fourth-power free).

nonn,fini,hard

T. D. Noe, Apr 06 2018