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 A133485 Integers n such that the polynomial x^(2n+2)+x+1 is primitive and irreducible over GF(2). 0
 0, 1, 2, 10, 29, 265, 449, 682 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS An integer n>1 belongs to this sequence iff A100730(n)=2^(2n+3)-2. Also, an integer n belongs to this sequence iff 2n+2 belongs to A073639. The polynomial x^(2n+2)+x+1 in the definition can be replaced by its reciprocal x^(2n+2)+x^(2n+1)+1. (2*a(n)+2) is a subsequence of A002475. - Manfred Scheucher, Aug 17 2015 a(9) >= (A002475(29)-2)/2 = 5098. LINKS Table of n, a(n) for n=1..8. MAPLE 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 PROG (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); 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 (Sage) def is_primitive(p): d = 2^(p.degree())-1 if not p.divides(x^d-1): return False for k in (d//q for q in d.prime_factors()): if p.divides(x^k-1): return False return True P. = GF(2)[] for n in range(1, 1000): p = x^(2*n+2)+x+1 if p.is_irreducible() and is_primitive(p): print(n) # Modification of the A002475 Script by Ruperto Corso # Manfred Scheucher, Aug 17 2015 CROSSREFS Cf. A002475, A073639, A100730. Sequence in context: A190186 A032250 A215790 * A215954 A098425 A098408 Adjacent sequences: A133482 A133483 A133484 * A133486 A133487 A133488 KEYWORD nonn,more,hard AUTHOR Max Alekseyev, Dec 02 2007, Feb 15 2008 EXTENSIONS a(2)=1 inserted by Michel Marcus, May 29 2013 STATUS approved

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Last modified October 1 14:44 EDT 2023. Contains 365826 sequences. (Running on oeis4.)