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 A054639 Queneau numbers: numbers n such that the Queneau-Daniel permutation {1, 2, 3, ..., n} -> {n, 1, n-1, 2, n-2, 3, ...} is of order n. 15
 1, 2, 3, 5, 6, 9, 11, 14, 18, 23, 26, 29, 30, 33, 35, 39, 41, 50, 51, 53, 65, 69, 74, 81, 83, 86, 89, 90, 95, 98, 99, 105, 113, 119, 131, 134, 135, 146, 155, 158, 173, 174, 179, 183, 186, 189, 191, 194, 209, 210, 221, 230, 231, 233, 239 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The troubadour Arnaut Daniel composed sestinas based on the permutation 123456 -> 615243, which cycles after 6 iterations. Roubaud quotes the number 141, but the corresponding Queneau-Daniel permutation is only of order 47 = 141/3. This appears to coincide with the numbers n such that a type-2 optimal normal basis exists for GF(2^n) over GF(2). But are these two sequences really the same? - Joerg Arndt, Feb 11 2008 The answer is Yes - see Theorem 2 of the Dumas reference. [Jean-Guillaume Dumas (Jean-Guillaume.Dumas(AT)imag.fr), Mar 20 2008] From Peter R. J. Asveld, Aug 17 2009: (Start) a(n) is the n-th T-prime (Twist prime). For N>=2, the family of twist permutations is defined by p(m,N) = +2m (mod 2N+1) if 1 <= m GroupTheory:-PermOrder(Perm(QD(n)))=n, [\$1..1000]); # Robert Israel, May 01 2016 MATHEMATICA a[p_] := Sum[Cos[2^n Pi/((2 p + 1) )], {n, 1, p}]; Select[Range, Reduce[a[#] == -1/2, Rationals] &] (* Gerry Martens, May 01 2016 *) PROG (PARI) is(n)= { if (n==1, return(1)); my( m=n%4 ); if ( m==4, return(0) ); my(p=2*n+1, r=znorder(Mod(2, p))); if ( !isprime(p), return(0) ); if ( m==3 && r==n, return(1) ); if ( r==2*n, return(1) ); \\ r == 1 or 2 return(0); } for(n=1, 10^3, if(is(n), print1(n, ", ")) ); \\ Joerg Arndt, May 02 2016 CROSSREFS Not to be confused with Queneau's "s-additive sequences", cf. A003044. Considered as sets A054639 is the union of A163782 (Josephus_2-primes) and A163781 (dual Josephus_2-primes); it also equals the union of A163777 (Archimedes_0-primes) and A163778 (Archimedes_1-primes). If b(n) and c(n) denote A071642 (shuffle primes) and A163776 (dual shuffle primes) respectively, then A054639 is the union of b(n)/2 and c(n)/2. - Peter R. J. Asveld, Aug 17 2009 Cf. A216371, A003558 (for which a(n) == n). Sequence in context: A225527 A008747 A327516 * A123399 A239010 A104738 Adjacent sequences: A054636 A054637 A054638 * A054640 A054641 A054642 KEYWORD nonn AUTHOR Gilles Esposito-Farese (gef(AT)cpt.univ-mrs.fr), May 17 2000 STATUS approved

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Last modified June 1 18:23 EDT 2023. Contains 363076 sequences. (Running on oeis4.)