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%I #23 Sep 16 2022 10:26:50
%S 4,6,12,22,28,36,46,52,60,70,78,100,102,148,166,172,180,190,196,198,
%T 238,262,268,270,292,310,316,348,358,366,372,382,388,420,460,462,478,
%U 486,502,508,540,556,598,606,612,646,652,660,676,700,708,718,742,750,756
%N a(n) is the n-th dS-prime (dual Shuffle prime).
%C 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.
%C 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).
%H P. R. J. Asveld, <a href="/A163776/b163776.txt">Table of n, a(n) for n=1..3612</a>
%H P. R. J. Asveld, <a href="http://dx.doi.org/10.1016/j.dam.2011.07.019">Permuting operations on strings and their relation to prime numbers</a>, Discrete Applied Mathematics 159 (2011) 1915-1932.
%H P. R. J. Asveld, <a href="http://eprints.eemcs.utwente.nl/20685/">Permuting operations on strings and the distribution of their prime numbers</a> (2011), TR-CTIT-11-24, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
%H P. R. J. Asveld, <a href="http://eprints.eemcs.utwente.nl/15678/">Some Families of Permutations and Their Primes </a> (2009), TR-CTIT-09-27, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
%H P. R. J. Asveld, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.216.1682">Permuting Operations on Strings-Their Permutations and Their Primes</a>, Twente University of Technology, 2014. <a href="http://doc.utwente.nl/67513">University link</a>.
%H <a href="/index/J#Josephus">Index entries for sequences related to the Josephus Problem</a>
%F a(n) = 2*A163781(n).
%e 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.
%Y 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).
%K nonn
%O 1,1
%A _Peter R. J. Asveld_, Aug 13 2009
%E a(33)-a(55) from _Andrew Howroyd_, Nov 11 2017
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