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?



Why do primes dislike dividing the sum of all the preceding primes?
Here is a heuristic argument that there is nothing to explain:


The probability that $p$ divides the sum of the preceding primes is $1/p$. So the expected number of primes less than $10^9$ with this property is $\sum\_{p \leq 10^9} \frac{1}{p}$. Using [Mertens' second theorem](https://en.wikipedia.org/wiki/Mertens%27_theorems),
$$\sum\_{p \leq 10^9} \frac{1}{p} \approx \log \log 10^9 + M \approx 3.3$$


Here $\log$ is natural log and $M \approx 0.26149$ is [Mertens' constant](https://en.wikipedia.org/wiki/Meissel%E2%80%93Mertens_constant). 


This is an example of the motto "$\log \log x$ goes to infinity but has never been observed to do so". It is quite common for people to look at primes $p$ which divide some quantity $a\_p$ and conclude that they are surprisingly rare when, in fact, they are simply growing as $\log \log N$ for the reason above.


