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A163781 a(n) is the n-th dJ_2 prime (dual Josephus_2 prime). 5
2, 3, 6, 11, 14, 18, 23, 26, 30, 35, 39, 50, 51, 74, 83, 86, 90, 95, 98, 99, 119, 131, 134, 135, 146, 155, 158, 174, 179, 183, 186, 191, 194, 210, 230, 231, 239, 243, 251, 254, 270, 278, 299, 303, 306, 323, 326, 330, 338, 350, 354, 359, 371, 375, 378 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The family of dual Josephus_2 (or dJ_2) permutations is defined by p(m,N)=(2N + 1 - F(m,2N + 1))/2 if 1<=m<=N, N>=2, where F(x,y) is the odd number such that 1<=F(x,y)<y and x=F(x,y)*(-2)^t (mod y) for the smallest t>=0. Note that F(2k + 1,y)=2k + 1 for 2k + 1<y, as t=0 applies. N is a dJ_2 prime if this permutation consists of a single cycle of length N.

dJ_2 permutations can also be defined using a numbering/elimination procedure similar to the definition of the Josephus_2 permutations in [R.L. Graham et al.], or in A163782; see [P. R. J. Asveld].

No formula is known for a(n): the dJ_2 primes have been found by exhaustive search. But we have: (1) N is dJ_2 prime iff p=2N+1 is a prime number and -2 generates Z_p^* (the multiplicative group of Z_p); (2) N is dJ_2 prime iff p=2N+1 is a prime number and exactly one of the following holds:

(a) N=2 (mod 4) and both +2 and -2 generate Z_p^*,

(b) N=3 (mod 4) and -2 generates Z_p^* but +2 does not.

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.

R. L. Graham, D. E. Knuth & O. Patashnik, Concrete Mathematics (1989), Addison-Wesley, Reading, MA. Sections 1.3 & 3.3.

LINKS

P. R. J. Asveld, Table of n, a(n) for n=1..6756

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

EXAMPLE

For N=6 we have

  m       | 1   2   3   4   5   6

  --------+----------------------

  F(m,13) | 1   7   3  11   5   9

  t       | 0   2   0   1   0   3

  p(m,6)  | 6   3   5   1   4   2

So the permutation is (1 6 2 3 5 4) and 6 is dJ_2 prime.

MATHEMATICA

okQ[n_] := Mod[n, 4] >= 2 && PrimeQ[2n+1] && MultiplicativeOrder[2, 2n+1] == If[OddQ[n], n, 2n];

Select[Range[1000], okQ] (* Jean-Fran├žois Alcover, Sep 23 2019, from PARI *)

PROG

(PARI)

ok(n)={n%4>=2 && isprime(2*n+1) && znorder(Mod(2, 2*n+1)) == if(n%2, n, 2*n)};

select(ok, [1..1000]) \\ Andrew Howroyd, Nov 11 2017

CROSSREFS

Considered as sets the union of A163781 and A163782 (J_2 primes) equals A054639 (T-primes or Queneau numbers), their intersection is equal to A163777 (Archimedes_0 primes). A163781 equals the union of A163777 and A163780 (Archimedes^-_1 primes).

Sequence in context: A294510 A218155 A042944 * A090304 A005211 A298702

Adjacent sequences:  A163778 A163779 A163780 * A163782 A163783 A163784

KEYWORD

nonn

AUTHOR

Peter R. J. Asveld, Aug 17 2009

EXTENSIONS

a(37)-a(55) from Andrew Howroyd, Nov 11 2017

STATUS

approved

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Last modified September 24 22:28 EDT 2020. Contains 337325 sequences. (Running on oeis4.)