A213052 Increasing sequence of primes p such that all of 2,3,5,...,prime(n) are primitive roots mod p.

3, 5, 53, 173, 293, 2477, 9173, 22613, 27653, 61613, 74093, 92333, 170957, 360293, 679733, 847997, 2004917, 69009533, 76553573, 138473837, 237536213, 777133013, 883597853, 1728061733, 2050312613, 5534091197, 9447241877, 49107823133, 65315700413

Offset: 1

Author

Joerg Arndt, Jun 03 2012

Keywords

• nonn
• hard

Comments

a(32) > 10^12. - Dana Jacobsen, Jul 13 2018

Links

• Dana Jacobsen, Table of n, a(n) for n = 1..31

Programs

PARI

N=10^10;
default(primelimit,N);
A=2;
{ forprime (p=3, N,
    q = 1;
    forprime (a=2, A,
        if ( znorder(Mod(a,p)) != p-1,  q=0; break() );
    );
    if ( q, A=nextprime(A+1); print1(p,", ") );
);}

Perl

use Math::Prime::Util ":all";
my($N,$A,$p,$a,@P7) = (10**11,2);
forprimes { $p=$_;
  if (   is_primitive_root(2,$p)
      && ($A < 3 || is_primitive_root(3,$p))
      && ($A < 5 || is_primitive_root(5,$p))
      && ($A < 7 || vecall { is_primitive_root($_,$p) } @P7)
    ) {
      print "$p\n";
      $A = next_prime($A);
      push @P7, $A if $A >= 7;
    }
} 3,$N;
# Dana Jacobsen, Jul 11 2018

Extensions

a(20)-a(27) from Joerg Arndt, Apr 10 2016

a(28)-a(29) from Dana Jacobsen, Jul 11 2018