A095303 Smallest number k such that k^n - 2 is prime.

4, 2, 9, 3, 3, 3, 7, 7, 3, 21, 9, 7, 19, 5, 7, 39, 15, 61, 15, 19, 21, 3, 19, 17, 21, 5, 21, 7, 85, 17, 7, 21, 511, 27, 27, 59, 3, 19, 91, 45, 3, 29, 321, 65, 9, 379, 69, 125, 49, 5, 9, 45, 289, 341, 61, 89, 171, 171, 139, 21, 139, 75, 25, 49, 15, 51, 57, 175, 31, 137, 147, 25, 441

Offset: 1

Author

Hugo Pfoertner, Jun 01 2004

Keywords

• nonn

Comments

The Bunyakovsky conjecture implies a(n) exists for all n. - Robert Israel, Jul 15 2018

Some of the results were computed using the PrimeFormGW (PFGW) primality-testing program. - Hugo Pfoertner, Nov 16 2019

Examples

			a(1) = 4 because 4^1 - 2 = 2 is prime, a(3) = 9 because 3^3 - 2 = 25, 5^3 - 2 = 123 and 7^3 - 2 = 341 = 11 * 31 are composite, whereas 9^3 - 2 = 727 is prime.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..600
• Wikipedia, Bunyakovsky conjecture

Crossrefs

Cf. A095304 (corresponding primes), A087576 (smallest k such that k^n+2 is prime), A095302 (corresponding primes).

Cf. A014224.

Programs

Maple

f:= proc(n) local k;
  for k from 3 by 2 do
    if isprime(k^n-2) then return k fi
  od
end proc:
f(1):= 4: f(2):= 2:
map(f, [$1..100]); # Robert Israel, Jul 15 2018

Mathematica

a095303[n_] := For[k = 1, True, k++, If[PrimeQ[k^n - 2], Return[k]]]; Array[a095303, 100] (* Jean-François Alcover, Mar 01 2019 *)

PARI

for (n=1,73,for(k=1,oo,if(isprime(k^n-2),print1(k,", ");break))) \\ Hugo Pfoertner, Oct 28 2018

Extensions

a(2) and a(46) corrected by T. D. Noe, Apr 03 2012