You might find "lattice of interpretability types of varieties" to be related. The literature on it is small enough (as well as the active group of researchers on the subject) that you can likely find connections from this area to your more general topic, or at the very least one of the researchers may supply you with a connection. However, the representation or intepretation is not necessarily a functor. If you need a categorical perspective, someone here may suggest Lawvere theories as an alternate direction of investigation.

I'm noticing that my weak counting estimate can be strengthened by noting that some odd composites in the interval must have three or more distinct prime factors. Also, the existence of such a smooth odd interval implies (by dividing by 3,5, etc.) smaller smooth intervals (although relatively less smooth since the composites involved are smaller. Do you know of an approach in the literature that uses observations like these?

Calculations using the stronger inequality yield $k \gt 1480$. Now odd multiples of 105 and even 1155 will appear, so this still yields a weak lower bound.

Prof. Kuperberg, the issue is not whether your question is good or bad; the issue is whether this is the place for it. MathOverflow has a purpose different from that of pursuing open problems: it is to get existing answers quickly to definite questions in a certain realm. Instead of presenting a chllenge, you might ask if there is recent progress, or if someone has tried a specific approach on a small part of the problem.

More refined estimates and some guesswork suggest $l \gt 100$ to satisfy the inequality involving the sum, and thus $k \gt 541$ and so $p \gt 10^{12}$.

Except that the interval is no longer prime free once $p \gt 2$. Although it has no odd primes when $p=3$. To be fair, 1 should also not be considered composite.

Slightly finer estimates suggest $k \gt 127$ (and $l \gt 40$). I am guessing the answer is always yes for the interesting case.

Likely yes, with an exception being the primes -2 and 2. (I disregard the simple cases where the consecutive primes differ by 6 or less.)

I think the number of subsemigroups of the full transformation semigroup on n labeled points exceeds by far the number of topologies on the same sized set. There should be many such examples without a topology.

If I read Igor Rivin's post correctly, then it seems Walter Feit proved for prime powers $g_n$ that they are large enough, and so that if I replace $f_n$ greatest prime factor with $f_n$ greatest prime power factor, this modified statement is a consequence of Feit's Theorem B.

I thought I could. A possible attack is to show f_n is really large for n prime, and then apply that fact recursively, but I think a proof by contradiction will be easier.