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A086758 a(n) is the smallest m such that the integer part of the first n powers of m^(1/n) are primes.

%I A086758 #24 Apr 07 2025 23:42:31
%S A086758 2,5,13,31,631,173,409,967,3450844193,39661481813,2076849234433,
%T A086758 52134281654579,14838980942616539,260230524377962793,
%U A086758 4563650703502319197,80032531899785490253,172111744128569095516889
%N A086758 a(n) is the smallest m such that the integer part of the first n powers of m^(1/n) are primes.
%C A086758 All terms of this sequence must be primes because floor((a(n)^(1/n))^n) = a(n).
%C A086758 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 A086758 R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see Exercise 1.75, p. 69.
%H A086758 Martin Raab, <a href="/A086758/b086758.txt">Table of n, a(n) for n = 1..53</a>
%H A086758 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 A086758 For[m=1, Union[Table[PrimeQ[Floor[Prime[m]^(k/n)]], {k, n}]]!={True}, m++ ]; Prime[m]
%e A086758 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.
%e A086758 a(8)=967 because the sequence {2, 5, 13, 31, 73, 173, 409, 967} consists entirely of primes, the i-th term in the sequence being floor(967^(i/8)) and 967 is the smallest integer with this property.
%t A086758 Do[Print[For[m=1, Union[Table[PrimeQ[Floor[Prime[m]^(k/n)]], {k, n}]]!={True}, m++ ]; Prime[m]], {n, 8}]
%Y A086758 Cf. A063636, A076255, A076357.
%K A086758 nonn
%O A086758 1,1
%A A086758 _Farideh Firoozbakht_, Aug 01 2003
%E A086758 Terms a(9) and following from _Jon E. Schoenfield_, May 15 2010