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A163776 a(n) is the n-th dS-prime (dual Shuffle prime). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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<N, p(N,N) = N if N is odd. N is dS-prime if p(m,N) consists of a single cycle of length N. So all dS-primes are even.

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).

REFERENCES

P. R. J. Asveld, Permuting Operations on Strings-Their Permutations and Their Primes, Twente University of Technology, 2014; http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.216.1682; http://doc.utwente.nl/67513/1/pospp.pdf.

LINKS

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.

Index entries for sequences related to the Josephus Problem

FORMULA

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

EXAMPLE

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.

CROSSREFS

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).

Sequence in context: A020141 A049478 A263458 * A050558 A331192 A255843

Adjacent sequences:  A163773 A163774 A163775 * A163777 A163778 A163779

KEYWORD

nonn

AUTHOR

Peter R. J. Asveld, Aug 13 2009

EXTENSIONS

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

STATUS

approved

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Last modified September 26 12:07 EDT 2020. Contains 337371 sequences. (Running on oeis4.)