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A371566 Primes p such that x^5 - x^4 - x^3 - x^2 - x - 1 is irreducible (mod p).

%I A371566 #14 Mar 28 2024 11:57:40
%S A371566 5,7,11,13,17,31,37,41,53,79,107,199,233,239,311,331,337,389,463,523,
%T A371566 541,547,557,563,577,677,769,853,937,971,1009,1021,1033,1049,1061,
%U A371566 1201,1237,1291,1307,1361,1427,1453,1543,1657,1699,1723,1747,1753,1759,1787,1801,1811,1861,1877,1997,1999
%N A371566 Primes p such that x^5 - x^4 - x^3 - x^2 - x - 1 is irreducible (mod p).
%H A371566 Robert Israel, <a href="/A371566/b371566.txt">Table of n, a(n) for n = 1..10000</a>
%p A371566 P:= x^5 - x^4 - x^3 - x^2 - x - 1:
%p A371566 select(p -> Irreduc(P) mod p, [seq(ithprime(i), i=1..1000)]); # _Robert Israel_, Mar 13 2024
%t A371566 P = x^5 - x^4 - x^3 - x^2 - x - 1;
%t A371566 Select[Prime[Range[1000]], IrreduciblePolynomialQ[P, Modulus -> #]&] (* _Jean-François Alcover_, Mar 24 2024, after _Robert Israel_ *)
%o A371566 (Python)
%o A371566 from itertools import islice
%o A371566 from sympy import Poly, nextprime
%o A371566 from sympy.abc import x
%o A371566 def A371566_gen(): # generator of terms
%o A371566     p = 2
%o A371566     while True:
%o A371566         if Poly(x*(x*(x*(x*(x-1)-1)-1)-1)-1, x, modulus=p).is_irreducible:
%o A371566             yield p
%o A371566         p = nextprime(p)
%o A371566 A371566_list = list(islice(A371566_gen(),20)) # _Chai Wah Wu_, Mar 14 2024
%o A371566 (PARI) a371566(upto) = forprime (p=2, upto, my(f=factormod(x^5 - x^4 - x^3 - x^2 - x - 1, p)); if(#f[,1]==1, print1(p,", "))) \\ _Hugo Pfoertner_, Mar 22 2024
%Y A371566 Contained in, but not equal to, A106309. Cf. A370830.
%K A371566 nonn
%O A371566 1,1
%A A371566 _Robert Israel_, Mar 27 2024