A346154 a(n) is the least prime of the form n^k+n+1 with k>1, or 0 if there is no such prime.

3, 7, 13, 0, 31, 43, 0, 73, 739, 0, 14653, 157, 0, 211, 241, 0, 307, 5851, 0, 421, 463, 0, 1801152661487, 601, 0, 457003, 757, 0, 24419, 27031, 0, 32801, 1123, 0, 144884079282928466796911, 1679653, 0, 1483, 59359, 0, 1723, 74131, 0, 85229, 8303765671, 0, 4879729, 110641, 0, 2551, 2334165173090503

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) = 739 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..67

Crossrefs

Cf. A346149.

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 n^i+n+1 fi od:
end proc:
f(1):= 3:
map(f, [$1..100]);

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 n**k + n + 1
print([a(n) for n in range(1, 52)]) # Michael S. Branicky, Jul 07 2021

Formula

a(n) = n^A346149(n) + n + 1 if A346149(n) > 0.