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A346156
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Primes of the form x^k+x+1 where k >= 2 and x >= 1.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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a(3) = 11 is a term because 11 = 2^3+2+1 and is prime.
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MAPLE
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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));
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PROG
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(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)))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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