

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
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OFFSET

1,2


COMMENTS

Previous name was: a(n) is the nth A^+_1prime (Archimedes^+_1 prime).
N is A^+_1prime 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) 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.
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] (* JeanFrançois 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^+_1primes are the T or Twistprimes congruent 1 (mod 4), these Tprimes are equal to the Queneaunumbers (A054639). For the related A_0, A_1 and A^_1primes, 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_2primes).
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



