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

2, 5, 13, 31, 631, 173, 409, 967, 3450844193, 39661481813, 2076849234433, 52134281654579, 14838980942616539, 260230524377962793, 4563650703502319197, 80032531899785490253, 172111744128569095516889

Offset

1,1

Comments

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

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.

References

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

Links

Martin Raab, Table of n, a(n) for n = 1..53

Carlos Rivera, Puzzle 227. Research Problem 1.75, Prime Puzzles and Problems Connection.

Formula

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

Example

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.

Mathematica

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

Crossrefs

Sequence in context: A378111 A082733 A095134 * A179257 A116702 A098156

Adjacent sequences: A086755 A086756 A086757 * A086759 A086760 A086761

Keyword

nonn

Author

Farideh Firoozbakht, Aug 01 2003

Extensions

Terms a(9) and following from Jon E. Schoenfield, May 15 2010

Status

approved