Powers such that the interval to the next power contains exactly two primes.
4, 27, 324, 512, 1331, 2197, 3364, 4900, 140608, 175561, 6436343, 14348907, 143384152904
1
No other terms less than 10^16. Is this sequence finite?
T. D. Noe, Plot of 13 terms
(Mma) PrimeCountQ[n1_, n2_, k_] := Module[{n = n1 + 1, cnt = 0}, While[n < n2 && cnt < k + 1, If[PrimeQ[n], cnt++]; n++]; cnt == k && n == n2]; nn = 10^12; lim = Log[2, nn]; ps = Select[Range[lim], PrimeQ]; t = Table[Table[{i^n, n}, {i, 2, nn^(1/n)}], {n, ps}]; t = Sort[Flatten[t, 1]]; lst = {}; Do[If[t[[n, 2]] > 2 || t[[n + 1, 2]] > 2, If[PrimeCountQ[t[[n, 1]], t[[n + 1, 1]], 2], AppendTo[lst, {t[[n, 1]], t[[n + 1, 1]]}]]], {n, Length[t] - 1}]; First /@ lst
Cf. A116086, A116455 (similar, but no primes), A178700 (single primes between powers).
Cf. S000142-S000151 (power intervals containing 1 to 5 primes)
nonn,more
T. D. Noe, Jul 17 2014