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 A071642 Numbers n such that x^n + x^(n-1) + x^(n-2) + ... + x + 1 is irreducible over GF(2). 18
 0, 1, 2, 4, 10, 12, 18, 28, 36, 52, 58, 60, 66, 82, 100, 106, 130, 138, 148, 162, 172, 178, 180, 196, 210, 226, 268, 292, 316, 346, 348, 372, 378, 388, 418, 420, 442, 460, 466, 490, 508, 522, 540, 546, 556, 562, 586, 612, 618, 652, 658, 660, 676, 700, 708, 756, 772 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS All such polynomials of odd degree > 1 are reducible over GF(2). For n >= 2, a(n) = A001122(n-2) - 1 due to the relationship between cycles and irreducibility. - T. D. Noe, Sep 09 2003 n such that a type-1 optimal normal basis of GF(2^n) (over GF(2)) exists. The corresponding field polynomial is the all-ones polynomial x^n+x^(n-1)+...+1. - Joerg Arndt, Feb 25 2008 From Peter R. J. Asveld, Aug 13 2009: (Start) a(n) is also the n-th S-prime (Shuffle prime) For N>=2, the family of shuffle permutations is defined by p(m,N) = 2m (mod N+1) if N is even, p(m,N) = 2m (mod N) if N is odd and 1<=m= 2 are numbers n such that P^n + P^(n-1) + P^(n-2) + ... + P + 1 is irreducible over GF(2), where P=x^2+x+1. - Luis H. Gallardo, Dec 23 2019 LINKS P. R. J. Asveld, Table of n, a(n) for n = 1..3605 Joerg Arndt, Matters Computational (The Fxtbook), section 42.9 "Gaussian normal bases", pp.914-920 P. R. J. Asveld, Permuting operations on strings and their relation to prime numbers, Discrete Applied Mathematics 159 (2011) 1915-1932. P. R. J. Asveld, Permuting operations on strings and the distribution of their prime numbers, (2011), TR-CTIT-11-24, Dept. of CS, Twente University of Technology, Enschede, The Netherlands. P. R. J. Asveld, Some Families of Permutations and Their Primes (2009), TR-CTIT-09-27, Dept. of CS, Twente University of Technology, Enschede, The Netherlands. P. R. J. Asveld, Permuting Operations on Strings-Their Permutations and Their Primes, Twente University of Technology, 2014. H. Caure, C. Agon, M. Andreatta, Modulus p Rhythmic Tiling Canons and some implementations in Open Music visual programming language, in Proceedings ICMC|SMC|2014 14-20 September 2014, Athens, Greece. M. Olofsson, VLSI Aspects on Inversion in Finite Fields, Dissertation No. 731, Dept. Elect. Engin., Linkoping, Sweden, 2002. Eric Weisstein's World of Mathematics, Irreducible Polynomial EXAMPLE For n=4 and n=6 we obtain the permutations (1 2 4 3) and (1 2 4)(3 6 5): 4 is S-prime, but 6 is not. [Peter R. J. Asveld, Aug 13 2009] MATHEMATICA Do[s = Sum[x^i, {i, 0, n}]; If[ ToString[ Factor[s, Modulus -> 2]] == ToString[s], Print[n]], {n, 2, 1000, 2}] Join[{0, 1}, Reap[For[p = 2, p < 10^3, p = NextPrime[p], If[ MultiplicativeOrder[2, p] == p-1, Sow[p-1]]]][[2, 1]]] (* Jean-François Alcover, Dec 10 2015, adapted from PARI *) PROG (PARI) forprime(p=3, 1000, if(znorder(Mod(2, p))==p-1, print1(p-1, ", "))) /* Joerg Arndt, Jul 05 2011 */ CROSSREFS Cf. A001122 (primes with primitive root 2). Sequence in context: A047463 A107059 A160716 * A226827 A266538 A265223 Adjacent sequences:  A071639 A071640 A071641 * A071643 A071644 A071645 KEYWORD easy,nonn AUTHOR N. J. A. Sloane, Jun 22 2002 EXTENSIONS Extended by Robert G. Wilson v, Jun 24 2002 Initial terms of b-file corrected by N. J. A. Sloane, Aug 31 2009 STATUS approved

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Last modified April 22 07:26 EDT 2021. Contains 343163 sequences. (Running on oeis4.)