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A350121 Increasing sequence of primes p == 3 (mod 4) such that all of 2,3,5,...,prime(n) are primitive roots mod p.

%I A350121 #37 May 14 2025 00:03:30
%S A350121 3,19,907,1747,2083,101467,350443,916507,1014787,6603283,27068563,
%T A350121 45287587,226432243,243060283,3946895803,5571195667,9259384843,
%U A350121 19633449763,229012273627,965558895907,2793054173947,5142304754563
%N A350121 Increasing sequence of primes p == 3 (mod 4) such that all of 2,3,5,...,prime(n) are primitive roots mod p.
%C A350121 It is possible, although rather unlikely, that any primes congruent to 3 (mod 4) will appear in A213052.
%C A350121 a(19) > 10^11.
%e A350121 a(2) = 19 since 19 is the smallest prime (congruent to 3 (mod 4)) such that the first two primes (2 and 3) are primitive roots.
%t A350121 max=0;Do[n=Prime@i;If[Mod[n,4]==3,k=1;While[MultiplicativeOrder[Prime@k,n]==n-1,k++];If[k-1>max,Print@n;max++]],{i,10^6}] (* _Giorgos Kalogeropoulos_, Dec 17 2021 *)
%o A350121 (PARI)
%o A350121 N=10^10;
%o A350121 default(primelimit, N);
%o A350121 A=2;
%o A350121 { forprime (p=3, N,
%o A350121     if (p%4==3,
%o A350121     q = 1;
%o A350121     forprime (a=2, A,
%o A350121         if ( znorder(Mod(a, p)) != p-1,  q=0; break() );
%o A350121     );
%o A350121     if ( q, A=nextprime(A+1); print1(p, ", ") );
%o A350121     );
%o A350121 ); }
%Y A350121 Cf. A213052.
%K A350121 nonn,more
%O A350121 1,1
%A A350121 _Paul Vanderveen_, Dec 15 2021
%E A350121 a(19) from _Daniel Suteu_, Dec 20 2021
%E A350121 a(20)-a(21) from _Paul Vanderveen_, May 08 2025