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A163781
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a(n) is the n-th dJ_2 prime (dual Josephus_2 prime).
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6
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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
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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R. L. Graham, D. E. Knuth & O. Patashnik, Concrete Mathematics (1989), Addison-Wesley, Reading, MA. Sections 1.3 & 3.3.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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okQ[n_] := Mod[n, 4] >= 2 && PrimeQ[2n+1] && MultiplicativeOrder[2, 2n+1] == If[OddQ[n], n, 2n];
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PROG
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(PARI)
ok(n)={n%4>=2 && isprime(2*n+1) && znorder(Mod(2, 2*n+1)) == if(n%2, n, 2*n)};
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CROSSREFS
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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).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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