TITLE: Prime factors of $\sum_{k=1}^{30}k^{k^k}$
QUESTION [11 upvotes]: I checked the prime factors of
$$\sum_{k=1}^{30}k^{k^k}$$
and did not find any upto $10^8$
Are there any useful restrictions to accelerate the search ?

REPLY [2 votes]: I wrote a small C program using the n_powmod2_preinv function in the Flint Number Theory Library. I used the tricks you already mentioned and checked up to $5\cdot10^{10}$ and didn't find any factors.