A346149 a(n) is the least integer k > 1 such that n^k + n + 1 is prime, or 0 if there is no such k.

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, 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, 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

Offset: 1

Author

J. M. Bergot and Robert Israel, Jul 07 2021

Keywords

• nonn

Comments

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

Conjecture: a(n) > 0 otherwise.

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 1..212

Programs

Maple

f:= proc(n) local i;
if n mod 3 = 1 then return 0 fi;
for i from 2 do if isprime(n^i+n+1) then return i fi od:
end proc:
f(1):= 2:
map(f, [$1..100]);

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

Python

from sympy import isprime
def a(n):
    if n > 1 and n%3 == 1: return 0
    k = 2
    while not isprime(n**k + n + 1): k += 1
    return k
print([a(n) for n in range(1, 96)]) # Michael S. Branicky, Jul 08 2021