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A163779 Numbers k of the form 4*j + 1 such that 2*k + 1 is a prime with primitive root 2. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

FORMULA

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

MATHEMATICA

okQ[n_] := Mod[n, 4] == 1 && PrimeQ[2n+1] && MultiplicativeOrder[2, 2n+1] == 2n;

Select[Range[1000], okQ] (* Jean-Fran├žois Alcover, Jun 30 2018, after Andrew Howroyd *)

PROG

(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

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).

Sequence in context: A147230 A192914 A303676 * A280487 A191013 A193487

Adjacent sequences:  A163776 A163777 A163778 * A163780 A163781 A163782

KEYWORD

nonn

AUTHOR

Peter R. J. Asveld, Aug 12 2009

EXTENSIONS

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

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

STATUS

approved

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Last modified September 29 10:54 EDT 2020. Contains 337428 sequences. (Running on oeis4.)