A163779 Numbers k of the form 4*j + 1 such that 2*k + 1 is a prime with primitive root 2.

1, 5, 9, 29, 33, 41, 53, 65, 69, 81, 89, 105, 113, 173, 189, 209, 221, 233, 245, 261, 273, 281, 293, 309, 329, 393, 413, 429, 441, 453, 473, 509, 545, 561, 585, 593, 629, 641, 645, 653, 713, 725, 741, 749, 761, 765, 785, 809, 833, 873, 893, 933, 953, 965, 989, 993

Offset: 1

Author

Peter R. J. Asveld, Aug 12 2009

Keywords

• nonn

Comments

Previous name was: a(n) is the n-th A^+_1-prime (Archimedes^+_1 prime).

N is A^+_1-prime iff N=1 (mod 4), p=2N+1 is a prime number and +2 generates Z_p^* (the multiplicative group of Z_p) but -2 does not.

Links

• Joerg Arndt, Table of n, a(n) for n = 1..10000 (first 3328 from P. R. J. Asveld)
• 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.
• P. Michael Hutchins, Reworded Definition

Crossrefs

The A^+_1-primes are the T- or Twist-primes congruent 1 (mod 4), these T-primes are equal to the Queneau-numbers (A054639). For the related A_0-, A_1- and A^-_1-primes, see A163777, A163778 and A163780. Considered as sets the union of A163779 and A163780 equals A163778, the union of A163779 and A163777 is equal to A163782 (J_2-primes).

Programs

Mathematica

okQ[n_] := Mod[n, 4] == 1 && PrimeQ[2n+1] && MultiplicativeOrder[2, 2n+1] == 2n;
Select[Range[1000], okQ] (* Jean-François Alcover, Jun 30 2018, after Andrew Howroyd *)

PARI

ok(n) = n%4==1 && isprime(2*n+1) && znorder(Mod(2, 2*n+1))==2*n;
select(ok, [1..1000]) \\ Andrew Howroyd, Nov 11 2017

Formula

2 * a(n) + 1 = A213051(n+1). - Joerg Arndt, Mar 23 2018

Extensions

a(32)-a(55) from Andrew Howroyd, Nov 11 2017

Term 1 prepended and new name from Joerg Arndt, Mar 23 2018