A163777 Even terms in the sequence of Queneau numbers A054639.

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

Offset

1,1

Comments

Previous name was: a(n) is the n-th A_0-prime (Archimedes_0 prime).

We have: (1) N is A_0-prime 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_0-prime 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), 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. 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] (* Jean-François 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((j-1)/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_0-primes are the even T- or Twist-primes, these T-primes are equal to the Queneau-numbers (A054639). For the related A_1-, A^+_1- and A^-_1-primes, see A163778, A163779 and A163780. Considered as sets A163777 is the intersection of the Josephus_2-primes (A163782) and the dual Josephus_2-primes (A163781), it also equals the difference of A054639 and the A_1-primes (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