A076357 -id:A076357 - cp's OEIS frontend

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

Farideh Firoozbakht, Aug 01 2003

nonn

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.

			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.
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.

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

Martin Raab, Table of n, a(n) for n = 1..53
Carlos Rivera, Puzzle 227. Research Problem 1.75, Prime Puzzles and Problems Connection.

Cf. A063636, A076255, A076357.

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

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

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

A094106 a(n) is the maximal length L of a "power floor prime" sequence, i.e., a sequence of the form floor(x^k), k = 1, 2, ..., L such that floor(x) = prime(n).

Original entry on oeis.org

8, 7, 8, 5, 10, 12, 16, 14, 18, 22, 24, 26, 27, 28, 34, 35, 37, 39, 40, 45, 43, 46, 49, 51, 55, 57

Offset: 1

Johann Wiesenbauer (j.wiesenbauer(AT)tuwien.ac.at), May 02 2004

			a(1)=8 because for x=111/47 the sequence [x^k], k=1,2,... 2,5,13,31,73,173,409,967,... starts with 8 primes and this is the maximum for any x with [x]=2. (Compare also A063636, though the rational number x = 1287/545 used there is not of minimal height!)

Crandall and Pomerance, "Prime numbers, a computational perspective", p. 69, Research Problem 1.75.

C. Rivera, Problem 42
C. Rivera, Puzzle 227
Eric Weisstein's World of Mathematics, Power Floor Prime Sequence

Cf. A076255, A076357.

a(22) = 46 from Johann Wiesenbauer (j.wiesenbauer(AT)tuwien.ac.at), Jun 03 2004
a(23) = 49 from Johann Wiesenbauer (j.wiesenbauer(AT)tuwien.ac.at), Jun 27 2004
a(24) = 51 from Johann Wiesenbauer (j.wiesenbauer(AT)tuwien.ac.at), Aug 08 2004
a(25) and a(26) from Michael Kenn (michael.kenn(AT)philips.com), Jan 03 2006, who says: To achieve this result I used a shared network of 37 computers over the Christmas holidays. The total calculation time was equivalent to slightly more than 1 CPU year of a P4 - 2.4GHz.

cp's OEIS Frontend

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

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