A178700 Lonely primes between two consecutive nontrivial powers.

127, 223, 251, 64007, 1295033, 2535525373, 542939080319

Offset: 1

Author

Marot Alain (marot.alain(AT)orange.fr), Jun 05 2010

Keywords

• nonn
• more

Comments

While the number of perfect powers < n is ~ sqrt(n) and the number of primes < n is ~ n/log(n), this does not preclude more terms from existing, but it does make it very unlikely. [Robert G. Wilson v, Jun 10 2010]

Examples

			5^3 < 127 < 2^7;
6^3 < 223 < 15^2;
3^5 < 251 < 2^8;
40^3 < 64007 < 253^2;
109^3 < 1295033 < 1138^2.
From _Carlos Rivera_, Nov 26 2013: (Start)
50354^2 < 2535525373 < 76^5;
8158^3 < 542939080319 < 736844^2. (End)
		

Programs

Mathematica

nextPerfectPower[n_] := Block[{k = n + 1}, While[ GCD @@ Last /@ FactorInteger@k == 1, k++ ]; k]; a = 1; b = 3; lst = {}; While[a < 15 10^8, If[ PrimePi@b == 1 + PrimePi@a, p = NextPrime@a; AppendTo[lst, p]; Print@p]; a = b; b = nextPerfectPower@b]; lst (* Robert G. Wilson v, Jun 10 2010 *)

Extensions

First 5 terms checked by Robert G. Wilson v, Jun 10 2010

a(6)-a(7) from Carlos Rivera, Nov 26 2013