1,2

The troubadour Arnaut Daniel composed sestinas based on the permutation 123456 -> 615243, which cycles after 6 iterations.

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. Comment from Jean-Guillaume Dumas (Jean-Guillaume.Dumas(AT)imag.fr), Mar 20 2008: The asnwer is Yes - see Theorem 2 of the Dumas reference.

Contribution from P. R. J. Asveld (infprja(AT)cs.utwente.nl), 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<k=ceiling((N+1)/2),

p(m,N) = -2m (mod 2N+1) if k<=m<N.

N is T-prime if p(m,N) consists of a single cycle of length N.

The twist permutation is the inverse of the Queneau-Daniel permutation.

N is T-prime iff p=2N+1 is a prime number and exactly one of the following three conditions holds;

(1) N=1 (mod 4) and +2 generates Z_p^* (the multiplicative group of Z_p) but -2 does not,

(2) N=2 (mod 4) and both +2 and -2 generate Z_p^*,

(3) N=3 (mod 4) and -2 generate Z_p^* bur +2 does not. (End)

M. Bringer, Sur un probleme de R. Queneau, Math. Sci. Humaines No 25 (1969) 13-20.

Jacques Roubaud, Bibliotheque Oulipienne No 65 (1992) and 66 (1993).

P.R.J. Asveld, Table of n, a(n) for n=1..10085

Joerg Arndt, draft of the fxtbook

Jean-Guillaume Dumas, Caracterisation des Quenines et leur representation spirale

G. Esposito-Farese, C program

P. R. J. Asveld, Permuting Operations on Strings --- Their Permutations and Their Primes (2009), TR-CTIT-09-26, Dept. of CS, Twente University of Technology, Enschede, The Netherlands. [From P. R. J. Asveld (infprja(AT)cs.utwente.nl), Aug 17 2009]

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. [From P. R. J. Asveld (infprja(AT)cs.utwente.nl), Aug 17 2009]

For N=6 and N=7 we obtain the permutations (1 2 4 5 3 6) and (1 2 4 7)(3 6)(5): 6 is T-prime, but 7 is not. [From P. R. J. Asveld (infprja(AT)cs.utwente.nl), Aug 17 2009]

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. [From P. R. J. Asveld (infprja(AT)cs.utwente.nl), Aug 17 2009]

Sequence in context: A102825 A070991 A008747 this_sequence A070757 A123399 A104738

Adjacent sequences: A054636 A054637 A054638 this_sequence A054640 A054641 A054642

nonn

Gilles Esposito-Farese (gef(AT)cpt.univ-mrs.fr), May 17 2000

Roubaud quotes the number 141, but the corresponding Queneau-Daniel permutation is only of order 47 = 141/3.