Thanks for moving the post.
Quote:
The rarity of these numbers might put the next beyond the powers of BatalovPropper.

Who knows, it's not like where GIMPS is currently at where 20M+ consecutive candidates are composite  at least I don't think so.
I always forget the estimate for the digit size of primorials but the FFT size remains very managable even up to 20,000,000# where it's 3M.
So if anyone was willing to put some larger ressources towards primorials or factorials, I'm pretty sure it'll yield some nice results before ending up in GIMPS waters.