TITLE: Why do primes dislike dividing the sum of all the preceding primes?
QUESTION [35 upvotes]: 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?

REPLY [24 votes]: Question Q1 seems to me an extremely hard problem, but I also believe that the answer is affirmative because the heuristic argument exposed by David Speyer.
I studied with FLorian Luca [1] a related problem that could help to answer question Q2:
Let  $A$ the set of integers $n$ such that the sum of the first n primes is divisible by $n$ (instead of $p_n$). In other words, $A$ is the set of $n$ such that the arithmetic mean of the first $n$ primes is an integer:
$A=\{ n: \frac{p_1+\cdots +p_n}n \in \mathbf Z  \}=\{ 1, 23, 53, 853, 11869, 117267, 339615, 3600489,..\}   $
These numbers are not so rare because in this case the heuristic gives $ A(x)\asymp \sum_{n\le x}\frac 1n\sim \log x$.
We could not prove that $A$ has infinite elements but we proved that they are rare:
$$A(x)\ll x e^{-C(\log x)^{3/5}(\log \log x)^{-1/5}}.$$
Later Matomaki [2]  proved the stronger estimate,
$A(x)\ll x^{\frac{19}{24}+\epsilon}$.
[1] Cilleruelo, Javier; Luca, Florian, On the sum of the first n primes. Q. J. Math. 59 (2008), no. 4. http://www.uam.es/personal_pdi/ciencias/cillerue/articulos.html
[2] Matomäki, Kaisa, A note on the sum of the first n primes. Q. J. Math. 61 (2010), no. 1,