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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.

%I A163779 #46 Jun 30 2018 02:18:58
%S A163779 1,5,9,29,33,41,53,65,69,81,89,105,113,173,189,209,221,233,245,261,
%T A163779 273,281,293,309,329,393,413,429,441,453,473,509,545,561,585,593,629,
%U A163779 641,645,653,713,725,741,749,761,765,785,809,833,873,893,933,953,965,989,993
%N A163779 Numbers k of the form 4*j + 1 such that 2*k + 1 is a prime with primitive root 2.
%C A163779 Previous name was: a(n) is the n-th A^+_1-prime (Archimedes^+_1 prime).
%C A163779 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.
%H A163779 Joerg Arndt, <a href="/A163779/b163779.txt">Table of n, a(n) for n = 1..10000</a> (first 3328 from P. R. J. Asveld)
%H A163779 P. R. J. Asveld, <a href="http://dx.doi.org/10.1016/j.dam.2011.07.019">Permuting operations on strings and their relation to prime numbers</a>, Discrete Applied Mathematics 159 (2011) 1915-1932.
%H A163779 P. R. J. Asveld, <a href="http://eprints.eemcs.utwente.nl/20685/">Permuting operations on strings and the distribution of their prime numbers</a> (2011), TR-CTIT-11-24, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
%H A163779 P. R. J. Asveld, <a href="http://eprints.eemcs.utwente.nl/15678/">Some Families of Permutations and Their Primes </a> (2009), TR-CTIT-09-27, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
%H A163779 P. R. J. Asveld, <a href="http://doc.utwente.nl/67513/">Permuting Operations on Strings-Their Permutations and Their Primes</a>, Twente University of Technology, 2014.
%H A163779 P. Michael Hutchins, <a href="/A163779/a163779.txt">Reworded Definition</a>
%F A163779 2 * a(n) + 1 = A213051(n+1). - _Joerg Arndt_, Mar 23 2018
%t A163779 okQ[n_] := Mod[n, 4] == 1 && PrimeQ[2n+1] && MultiplicativeOrder[2, 2n+1] == 2n;
%t A163779 Select[Range[1000], okQ] (* _Jean-François Alcover_, Jun 30 2018, after _Andrew Howroyd_ *)
%o A163779 (PARI)
%o A163779 ok(n) = n%4==1 && isprime(2*n+1) && znorder(Mod(2, 2*n+1))==2*n;
%o A163779 select(ok, [1..1000]) \\ _Andrew Howroyd_, Nov 11 2017
%Y A163779 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).
%K A163779 nonn
%O A163779 1,2
%A A163779 _Peter R. J. Asveld_, Aug 12 2009
%E A163779 a(32)-a(55) from _Andrew Howroyd_, Nov 11 2017
%E A163779 Term 1 prepended and new name from _Joerg Arndt_, Mar 23 2018