Prove that a prime number $p$ divides  $1^1+2^2+3^3+....+(p-1)^{p-1}$ if and only if $p$ = 17 or 19.

Consider the question whether it is true that a prime number $p$ divides  $1^1+2^2+3^3+....+(p-1)^{p-1}$ if and only if $p \in \{17,19\}$.

For the obvious heuristic reasons, for large $n$ one would expect there to be roughly $\ln(\ln(n))$ such primes $p < n$, however it seems that presently no examples other than 17 and 19 are known.

Is there a more efficient way of looking for examples than the brute force method of testing the primes one-by-one?