A346156 Primes of the form x^k+x+1 where k >= 2 and x >= 1.

3, 7, 11, 13, 19, 31, 43, 67, 73, 131, 157, 211, 223, 241, 307, 421, 463, 521, 601, 631, 733, 739, 757, 1123, 1303, 1483, 1723, 1741, 2551, 2971, 3307, 3391, 3541, 3907, 4099, 4423, 4831, 4931, 5113, 5701, 5851, 6007, 6163, 6481, 6571, 8011, 8191, 9283, 9901, 10303, 11131, 12211, 12433, 13807

Offset: 1

Author

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

Keywords

• nonn

Comments

Primes p such that p-1 is in A253913.

Primes with more than one representation of this form include 31 = 3^3+3+1 = 5^2+5+1 and 131 = 2^7+2+1 = 5^3+5+1. Are there any others?

There are no others with more than one representation (except 3, trivially) < 10^19 (first 170385840 terms). - Michael S. Branicky, Jul 08 2021

Examples

			a(3) = 11 is a term because 11 = 2^3+2+1 and is prime.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A253913, A346154.

Programs

Maple

N:= 10^8: # for terms <= N
S:= {3}:
for k from 2 to ilog2(N-1) do
  S:= S union select(t -> t<= N and isprime(t),{seq(x^k+x+1,x=2..floor(N^(1/k)))}):
od:
sort(convert(S,list));

Python

from sympy import isprime
def aupto(lim):
    xkx = set(x**k + x + 1 for k in range(2, lim.bit_length()) for x in range(int(lim**(1/k))+2))
    return sorted(filter(isprime, filter(lambda t: t<=lim, xkx)))
print(aupto(14000)) # Michael S. Branicky, Jul 07 2021