%I #13 Jul 09 2021 22:58:09

%S 2,2,2,0,2,2,0,2,3,0,4,2,0,2,2,0,2,3,0,2,2,0,9,2,0,4,2,0,3,3,0,3,2,0,

%T 15,4,0,2,3,0,2,3,0,3,6,0,4,3,0,2,9,0,3,2,0,3,2,0,2,3,0,2,73,0,12,2,0,

%U 595,2,0,2,4,0,3,2,0,2,2,0,2,7,0,3,30,0,21,3,0,2,2,0,7,67,0,3

%N a(n) is the least integer k > 1 such that n^k + n + 1 is prime, or 0 if there is no such k.

%C a(n) = 0 if n == 1 (mod 3) and n > 1.

%C Conjecture: a(n) > 0 otherwise.

%H Robert Israel, <a href="/A346149/b346149.txt">Table of n, a(n) for n = 1..212</a>

%e a(9) = 3 because 9^3 + 9 + 1 = 739 is prime while 9^2+9+1 is not.

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

%p if n mod 3 = 1 then return 0 fi;

%p for i from 2 do if isprime(n^i+n+1) then return i fi od:

%p end proc:

%p f(1):= 2:

%p map(f, [$1..100]);

%o (PARI) a(n) = if ((n>1) && ((n%3)==1), 0, my(k=2); while (!isprime(n^k+n+1), k++); k); \\ _Michel Marcus_, Jul 07 2021

%o (Python)

%o from sympy import isprime

%o def a(n):

%o if n > 1 and n%3 == 1: return 0

%o k = 2

%o while not isprime(n**k + n + 1): k += 1

%o return k

%o print([a(n) for n in range(1, 96)]) # _Michael S. Branicky_, Jul 08 2021

%K nonn

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Jul 07 2021