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.
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.
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