A095303 - OEIS

%I #37 Nov 16 2019 16:31:43

%S 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,

%T 85,17,7,21,511,27,27,59,3,19,91,45,3,29,321,65,9,379,69,125,49,5,9,

%U 45,289,341,61,89,171,171,139,21,139,75,25,49,15,51,57,175,31,137,147,25,441

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

%C The Bunyakovsky conjecture implies a(n) exists for all n. - _Robert Israel_, Jul 15 2018

%C Some of the results were computed using the PrimeFormGW (PFGW) primality-testing program. - _Hugo Pfoertner_, Nov 16 2019

%H Robert Israel, <a href="/A095303/b095303.txt">Table of n, a(n) for n = 1..600</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bunyakovsky_conjecture">Bunyakovsky conjecture</a>

%e 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.

%p f:= proc(n) local k;

%p   for k from 3 by 2 do

%p     if isprime(k^n-2) then return k fi

%p   od

%p end proc:

%p f(1):= 4: f(2):= 2:

%p map(f, [$1..100]); # _Robert Israel_, Jul 15 2018

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

%o (PARI) for (n=1,73,for(k=1,oo,if(isprime(k^n-2),print1(k,", ");break))) \\ _Hugo Pfoertner_, Oct 28 2018

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

%Y Cf. A014224.

%K nonn

%O 1,1

%A _Hugo Pfoertner_, Jun 01 2004

%E a(2) and a(46) corrected by _T. D. Noe_, Apr 03 2012