Probability of a prime $p=3\pmod 4$ occurring in A213052

As you may notice, A213052 contains primes mostly congruent to $$1\pmod 4$$ (in fact, all of the known ones are except $$3$$).

Consider the sequence of smallest primes $$p_n$$ such that $$2,3,5,7,11,13,...$$ (first $$n$$ primes) are all primitive roots $$\pmod {p_n}$$.

So clearly, primes $$p = 3\pmod 4$$ could be part of this sequence (it is fairly easy to construct such primes):

The smallest example of a prime $$p=3\pmod 4$$ where $$2, 3, 5$$ are primitive roots for example, is $$907$$, while without this restriction, it is $$53$$. This can be done for arbitrary first $$n$$ primes, with such a restriction, we have the following $$p_n$$ congruent to $$3 \pmod 4$$:

$$3, 19, 907, 1747, 2083, 101467, 350443,...$$

These are much larger than the original sequence:

$$3, 5, 53, 173, 293, 2477, 9173$$

So I suspect that if a prime congruent to $$3 \pmod 4$$ is in A213052, it is pretty rare, although how likely is it to occur?

• Do you mean that the first $n$ primes must be primitive roots ? And we look at the smallest prime with this property and then the smallest of the form $4k+3$ ? Dec 15, 2021 at 11:08
• Aditionally, if you mention how such a prime can be constructed this might help to find out why primes of the form $4n+1$ are preferred. Dec 15, 2021 at 11:50
• I think it has something to do with $3$ being a Quadratic Non-Residue but it could be something else too. Dec 15, 2021 at 18:13