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

Author

Peter R. J. Asveld, Aug 13 2009

Keywords

• nonn

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

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).

Examples

			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.
		

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.
• 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

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).

Formula

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

Extensions

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

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