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A163779
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Numbers k of the form 4*j + 1 such that 2*k + 1 is a prime with primitive root 2.
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6
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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
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1,2
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COMMENTS
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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.
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LINKS
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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
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FORMULA
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2 * a(n) + 1 = A213051(n+1). - Joerg Arndt, Mar 23 2018
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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Peter R. J. Asveld, Aug 12 2009
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EXTENSIONS
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a(32)-a(55) from Andrew Howroyd, Nov 11 2017
Term 1 prepended and new name from Joerg Arndt, Mar 23 2018
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STATUS
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approved
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