How rare are the primes $p$ such that $p$ divides the sum of all primes less than $p$?
The naive heuristic is not so naive, although maybe not quite true. Here is a graph of $S_n \mod p_n$ versus $p_n$ for the first 2000 primes.
It looks rather random except for the part from about 10000 to 15000 that shows an interesting pattern. A closer look in other places reveals similar patterns in other places too, e.g. here it is from 90000 to 104000:
Can anybody explain this effect?
The next one is 415074643. Apparently that's the largest one known. See https://oeis.org/A007506