I was investigating primes with the property that the sum of the first $n$ primes is divisible by $p_n$. It turns out that these primes are extremely extremely rare. For primes less than $10^9$, I have found that there are only five primes with this property:

$$ p_1 = 2 $$

$$ p_3 = 5 $$

$$ p_{20} = 71 $$

$$ p_{31464} = 369,119 $$

$$ p_{22096548} = 415,074,643 $$

This raises the curious and equivalent questions:

**Q1.** Are there infinitely many primes which divide the sum of all the preceding primes?

**Q2**. Even if we assume that there are infinitely many such primes, why are they so rare? In other words, why do primes dislike dividing the sum of all the preceding primes? Is there any heuristic argument to show that such primes will indeed be extremely rare?