cp's OEIS Frontend

A133485 Integers k such that the polynomial x^(2k+2) + x + 1 is primitive and irreducible over GF(2).

%I A133485 #41 Oct 14 2023 21:15:57
%S A133485 0,1,2,10,29,265,449,682
%N A133485 Integers k such that the polynomial x^(2k+2) + x + 1 is primitive and irreducible over GF(2).
%C A133485 An integer k > 1 belongs to this sequence iff A100730(k) = 2^(2k+3) - 2.
%C A133485 Also, an integer k belongs to this sequence iff 2k+2 belongs to A073639.
%C A133485 The polynomial x^(2k+2) + x + 1 in the definition can be replaced by its reciprocal x^(2k+2) + x^(2k+1) + 1.
%C A133485 (2*a(n)+2) is a subsequence of A002475. - _Manfred Scheucher_, Aug 17 2015
%C A133485 a(9) >= (A002475(29) - 2)/2 = 5098.
%p A133485 select(n -> (Irreduc(x^(2*n+2)+x+1) mod 2) and (Primitive(x^(2*n+2)+x+1) mod 2), [$0..500]); # _Robert Israel_, Aug 17 2015
%o A133485 (PARI) polisprimitive(poli)=np = 2^poldegree(poli)-1; if (type((x^np-1)/poli) != "t_POL", return (0)); forstep(k=np-1, 1, -1, if (type((x^k-1)/poli) == "t_POL", return (0));); return(1);
%o A133485 lista(nn) = {for (n=0, nn, poli = Mod(1,2)*(x^(2*n+2)+x+1); if (polisirreducible(poli) && polisprimitive(poli), print1(n, ", ")););} \\ _Michel Marcus_, May 27 2013
%o A133485 (Sage)
%o A133485 def is_primitive(p):
%o A133485     d = 2^(p.degree())-1
%o A133485     if not p.divides(x^d-1): return False
%o A133485     for k in (d//q for q in d.prime_factors()):
%o A133485         if p.divides(x^k-1): return False
%o A133485     return True
%o A133485 P.<x> = GF(2)[]
%o A133485 for n in range(1,1000):
%o A133485     p = x^(2*n+2)+x+1
%o A133485     if p.is_irreducible() and is_primitive(p):
%o A133485         print(n)
%o A133485 # Modification of the A002475 Script by _Ruperto Corso_
%o A133485 # _Manfred Scheucher_, Aug 17 2015
%Y A133485 Cf. A002475, A073639, A100730.
%K A133485 nonn,more,hard
%O A133485 1,3
%A A133485 _Max Alekseyev_, Dec 02 2007, Feb 15 2008
%E A133485 a(2)=1 inserted by _Michel Marcus_, May 29 2013