OFFSET

1,1

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

LINKS

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

EXAMPLE

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

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));

PROG

(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

CROSSREFS

KEYWORD

nonn

AUTHOR

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

STATUS

approved