

A163777


Even terms in the sequence of Queneau numbers A054639.


10



2, 6, 14, 18, 26, 30, 50, 74, 86, 90, 98, 134, 146, 158, 174, 186, 194, 210, 230, 254, 270, 278, 306, 326, 330, 338, 350, 354, 378, 386, 398, 410, 414, 426, 438, 470, 530, 554, 558, 606, 614, 618, 638, 650, 686, 690, 726, 746, 774, 810, 818, 834, 846, 866, 870
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

Previous name was: a(n) is the nth A_0prime (Archimedes_0 prime).
We have: (1) N is A_0prime if and only if N is even, p = 2N + 1 is a prime number and both +2 and 2 generate Z_p^* (the multiplicative group of Z_p); (2) N is A_0prime if and only if N = 2 (mod 4), p = 2N + 1 is a prime number and both +2 and 2 generate Z_p^*.


LINKS

P. R. J. Asveld, Table of n, a(n) for n = 1..3378.
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.
P. R. J. Asveld, Permuting Operations on StringsTheir Permutations and Their Primes, Twente University of Technology, 2014. Another link.
Index entries for sequences related to the Josephus Problem


FORMULA

a(n) = 2*A137310(n).  Andrew Howroyd, Nov 11 2017


MATHEMATICA

okQ[n_] := EvenQ[n] && PrimeQ[2n+1] && MultiplicativeOrder[2, 2n+1] == 2n;
Select[Range[1000], okQ] (* JeanFrançois Alcover, Sep 10 2019, from PARI *)


PROG

(PARI)
Follow(s, f)={my(t=f(s), k=1); while(t>s, k++; t=f(t)); if(s==t, k, 0)}
ok(n)={n>1 && n==Follow(1, j>ceil((n+1)/2)  (1)^j*ceil((j1)/2))}
select(ok, [1..1000]) \\ Andrew Howroyd, Nov 11 2017
(PARI)
ok(n)={n%2==0 && isprime(2*n+1) && znorder(Mod(2, 2*n+1)) == 2*n}
select(ok, [1..1000]) \\ Andrew Howroyd, Nov 11 2017


CROSSREFS

The A_0primes are the even T or Twistprimes, these Tprimes are equal to the Queneaunumbers (A054639). For the related A_1, A^+_1 and A^_1primes, see A163778, A163779 and A163780. Considered as sets A163777 is the intersection of the Josephus_2primes (A163782) and the dual Josephus_2primes (A163781), it also equals the difference of A054639 and the A_1primes (A163779).
Cf. A137310.
Sequence in context: A280082 A139269 A186299 * A215807 A140525 A189804
Adjacent sequences: A163774 A163775 A163776 * A163778 A163779 A163780


KEYWORD

nonn


AUTHOR

Peter R. J. Asveld, Aug 11 2009


EXTENSIONS

Definition simplified by Michel Marcus, May 27 2013
a(33)a(55) from Andrew Howroyd, Nov 11 2017
New name from Joerg Arndt, Mar 23 2018, edited by M. F. Hasler, Mar 24 2018


STATUS

approved



