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

1

`%I #21 Jul 13 2021 01:37:08
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`%S 3,7,11,13,19,31,43,67,73,131,157,211,223,241,307,421,463,521,601,631,
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`%T 733,739,757,1123,1303,1483,1723,1741,2551,2971,3307,3391,3541,3907,
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`%U 4099,4423,4831,4931,5113,5701,5851,6007,6163,6481,6571,8011,8191,9283,9901,10303,11131,12211,12433,13807
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`%N Primes of the form x^k+x+1 where k >= 2 and x >= 1.
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`%C Primes p such that p-1 is in A253913.
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`%C 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?
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`%C 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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`%H Robert Israel, <a href="/A346156/b346156.txt">Table of n, a(n) for n = 1..10000</a>
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`%e a(3) = 11 is a term because 11 = 2^3+2+1 and is prime.
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`%p N:= 10^8: # for terms <= N
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`%p S:= {3}:
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`%p for k from 2 to ilog2(N-1) do
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`%p S:= S union select(t -> t<= N and isprime(t),{seq(x^k+x+1,x=2..floor(N^(1/k)))}):
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`%p od:
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`%p sort(convert(S,list));
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`%o (Python)
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`%o from sympy import isprime
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`%o def aupto(lim):
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`%o xkx = set(x**k + x + 1 for k in range(2, lim.bit_length()) for x in range(int(lim**(1/k))+2))
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`%o return sorted(filter(isprime, filter(lambda t: t<=lim, xkx)))
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`%o print(aupto(14000)) # _Michael S. Branicky_, Jul 07 2021
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`%Y Cf. A253913, A346154.
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`%K nonn
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`%O 1,1
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`%A _J. M. Bergot_ and _Robert Israel_, Jul 07 2021
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