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

1,1

Comments

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

Conjecture: a(n) > 0 otherwise.

Links

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

Example

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

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]);

Prog

(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

Crossrefs

Sequence in context: A217864 A002100 A108352 * A215883 A277024 A317528

Adjacent sequences: A346146 A346147 A346148 * A346150 A346151 A346152

Keyword

nonn

Author

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

Status

approved