%I #20 Aug 24 2024 20:21:52

%S 2,5,13,31,631,173,409,967,3450844193,39661481813,2076849234433,

%T 52134281654579,14838980942616539,260230524377962793,

%U 4563650703502319197,80032531899785490253,172111744128569095516889

%N a(n) is the smallest m such that the integer part of the first n powers of m^(1/n) are primes.

%C All terms of this sequence must be primes because floor((a(n)^(1/n))^n) = a(n).

%C Floor[(a(8)^(1/8))^k] = floor[(1287/545)^k] for k=1..10 (see puzzle 227). If a(9) exists it must be greater than 22000000.

%D R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see Exercise 1.75, p. 69.

%H Martin Raab, <a href="/A086758/b086758.txt">Table of n, a(n) for n = 1..53</a>

%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_227.htm">Puzzle 227. Research Problem 1.75</a>, Prime Puzzles and Problems Connection.

%F For[m=1, Union[Table[PrimeQ[Floor[Prime[m]^(k/n)]], {k, n}]]!={True}, m++ ]; Prime[m]

%e a(5)=631 because floor(631^(1/5)) = 3, floor(631^(2/5)) = 13, floor(631^(3/5)) = 47, floor(631^(4/5)) = 173 and floor(631^(5/5)) = 631 are primes and 631 is the smallest m with this property.

%t Do[Print[For[m=1, Union[Table[PrimeQ[Floor[Prime[m]^(k/n)]], {k, n}]]!={True}, m++ ]; Prime[m]], {n, 8}]

%K nonn

%O 1,1

%A _Farideh Firoozbakht_, Aug 01 2003

%E Terms a(9) and following from _Jon E. Schoenfield_, May 15 2010