A213051 -id:A213051 - cp's OEIS frontend

A163779 Numbers k of the form 4j + 1 such that 2k + 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

Peter R. J. Asveld, Aug 12 2009

nonn

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.

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

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).

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 *)

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

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

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

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

A213050 Primes of the form 4*k+1 with primitive root +2.

Original entry on oeis.org

5, 13, 29, 37, 53, 61, 101, 149, 173, 181, 197, 269, 293, 317, 349, 373, 389, 421, 461, 509, 541, 557, 613, 653, 661, 677, 701, 709, 757, 773, 797, 821, 829, 853, 877, 941, 1061, 1109, 1117, 1213, 1229, 1237, 1277, 1301, 1373, 1381, 1453, 1493, 1549, 1621

Offset: 1

Joerg Arndt, Jun 03 2012

nonn

Primes p such that both +2 and -2 are primitive roots mod p.

A001122 is the union of A213050 and A213051.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A213051 (primes 4*k+3 with primitive root +2).

Cf. A001122 (primitive root +2), A007885 (primitive root +2 or -2).

Select[Prime[Range[300]], Mod[#, 4] == 1 && PrimitiveRoot[#, 2] == 2&] (* Jean-François Alcover, Jul 22 2018 *)

{ forprime (p=3, 10^4,
    rp = znorder(Mod(+2,p));
    rm = znorder(Mod(-2,p));
    if ( (rp==p-1) && (rm==p-1), print1(p,", ") );
);}

cp's OEIS Frontend

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

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A213050 Primes of the form 4*k+1 with primitive root +2.

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cp's OEIS Frontend

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

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PARI

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A213050 Primes of the form 4*k+1 with primitive root +2.

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Author

Keywords

Comments

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Programs

Mathematica

PARI

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