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A054639
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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.
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12
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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; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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.
The answer is Yes - see Theorem 2 of the Dumas reference. [Jean-Guillaume Dumas (Jean-Guillaume.Dumas(AT)imag.fr), Mar 20 2008]
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)
The sequence name says the permutation is of order n, but P. R. J. Asveld's comment says it's an n-cycle. Is there a proof that those conditions are equivalent for the Queneau-Daniel permutation? (They are not equivalent for any arbitrary permutation, e.g. (123)(45)(6) has order 6 but isn't a 6-cycle.) More generally, I have found that for all n <= 9450, (order of Queneau-Daniel permutation) = (length of orbit of 1) = A003558(n). Does this hold for all n? - David Wasserman, Aug 30 2011
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REFERENCES
| 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).
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LINKS
| P. R. J. Asveld, Table of n, a(n) for n = 1..10085
Joerg Arndt, 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, 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.
Jean-Guillaume Dumas, Caracterisation des Quenines et leur representation spirale
G. Esposito-Farese, C program
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EXAMPLE
| 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]
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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. [From P. R. J. Asveld (infprja(AT)cs.utwente.nl), Aug 17 2009]
Sequence in context: A102825 A070991 A008747 * A070757 A123399 A104738
Adjacent sequences: A054636 A054637 A054638 * A054640 A054641 A054642
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KEYWORD
| nonn
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AUTHOR
| Gilles Esposito-Farese (gef(AT)cpt.univ-mrs.fr), May 17 2000
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EXTENSIONS
| Roubaud quotes the number 141, but the corresponding Queneau-Daniel permutation is only of order 47 = 141/3.
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