A054639 -id:A054639 - cp's OEIS frontend

A163776 a(n) is the n-th dS-prime (dual Shuffle prime).

4, 6, 12, 22, 28, 36, 46, 52, 60, 70, 78, 100, 102, 148, 166, 172, 180, 190, 196, 198, 238, 262, 268, 270, 292, 310, 316, 348, 358, 366, 372, 382, 388, 420, 460, 462, 478, 486, 502, 508, 540, 556, 598, 606, 612, 646, 652, 660, 676, 700, 708, 718, 742, 750, 756

Offset: 1

Peter R. J. Asveld, Aug 13 2009

nonn

For N>=2, the family of dual 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

No formula is known for a(n): the dS-primes have been found by exhaustive search. But we have: N is dS-prime iff p=N+1 is an odd prime number and -2 generates Z_p^* (the multiplicative group of Z_p).

			For N=6 and N=10 we obtain the permutations (1 5 4 6 2 3) and (1 9 4 3 5)(2 7 8 6 10): 6 is dS-prime, but 10 is not.

P. R. J. Asveld, Table of n, a(n) for n=1..3612
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. University link.
Index entries for sequences related to the Josephus Problem

a(n)/2 results in the dual Josephus_2-primes (A163781). Considered as sets a(n)/2 is the union of A163777 and A163780. If b(n) denotes the shuffle primes (A071642), then the union of a(n)/2 and b(n)/2 is equal to the Twist-primes or Queneau numbers (A054639), their intersection is equal to the Archimedes_0-primes (A163777).

a(n) = 2*A163781(n).

a(33)-a(55) from Andrew Howroyd, Nov 11 2017

A123399 Orders of "Gray" fields.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 11, 14, 23, 26, 29, 30, 33, 35, 39, 41, 51, 53, 65, 69, 74, 81, 83, 86, 89, 90, 95, 105, 113, 119, 131, 135, 146, 155, 158, 173, 179, 183, 189, 191, 209, 210, 221, 230, 231, 233, 239, 243, 245, 251, 254, 261, 273, 281, 293, 299, 303, 306, 309, 323, 326, 329, 330, 359, 371, 375, 386, 398, 411, 413, 419

Offset: 1

N. J. A. Sloane, Oct 15 2006

nonn

Numbers n such that there is a type-2 optimal normal basis over GF(2) and the corresponding polynomial is primitive. Subsequence of A054639. [Joerg Arndt, Apr 28 2012]

D. E. Knuth, The Art of Computer Programming, Section 7.2.1.1, Problem 31.

Joerg Arndt, Table of n, a(n) for n = 1..133

Cf. A054639.

Let c_1(x) = x+1, c_2(x) = x^2+x+1, c_{j+1}(x) = x*c_j(x) + c_{j-1}(x) be polynomials over GF(2). Then n is in the sequence iff c_n(x) is a primitive irreducible polynomial.

Terms >=105 by Joerg Arndt, Apr 28 2012.

A136250 Numbers n such that optimal normal basis exists for GF(2^n) over GF(2).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 14, 18, 23, 26, 28, 29, 30, 33, 35, 36, 39, 41, 50, 51, 52, 53, 58, 60, 65, 66, 69, 74, 81, 82, 83, 86, 89, 90, 95, 98, 99, 100, 105, 106, 113, 119, 130, 131, 134, 135, 138, 146, 148, 155, 158, 162, 172, 173, 174, 178, 179, 180, 183, 186

Offset: 1

Joerg Arndt, Mar 17 2008

nonn

An optimal normal basis for GF(2^n) is either of type-1 (A071642) or type-2 (A054639).

Joerg Arndt, Mar 17 2008, Table of n, a(n) for n = 1..240
Joerg Arndt, Matters Computational (The Fxtbook)

Cf. A071642 and A054639.

Edited by N. J. A. Sloane, Apr 08 2008

A014109 Number of possible circular rhymes of n strophes.

Original entry on oeis.org

1, 2, 3, 5, 9, 11, 14, 18, 23, 26, 29, 30, 33, 35, 39, 41

Offset: 1

Simon Plouffe, Paul Simon (paulsimn(AT)mail.odyssee.net)

Adding 6 to the sequence results in the first few Queneau numbers A054639. The entries a(n) are the first few generalizations of a verse form called sextine (or sestina in Italian) excluding the original sextine which was based on the number 6. The Queneau numbers characterize all these generalizations (including the number 6). For references see A054639.

From the OULIPO group (including Raymond Queneau), see OULIPO, Atlas de Littérature Potentielle, Coll. Idées, Gallimard, 1981, pp. 432, esp. p. 49.

Cf. A054639.

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A163776 a(n) is the n-th dS-prime (dual Shuffle prime).

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A123399 Orders of "Gray" fields.

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A136250 Numbers n such that optimal normal basis exists for GF(2^n) over GF(2).

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A014109 Number of possible circular rhymes of n strophes.

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