

A163776


a(n) is the nth dSprime (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
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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 dSprime if p(m,N) consists of a single cycle of length N. So all dSprimes are even.
No formula is known for a(n): the dSprimes have been found by exhaustive search. But we have: N is dSprime 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 StringsTheir 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) 19151932.
P. R. J. Asveld, Permuting operations on strings and the distribution of their prime numbers (2011), TRCTIT1124, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Some Families of Permutations and Their Primes (2009), TRCTIT0927, 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 dSprime, but 10 is not.


CROSSREFS

a(n)/2 results in the dual Josephus_2primes (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 Twistprimes or Queneau numbers (A054639), their intersection is equal to the Archimedes_0primes (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



