Are there functions or algorithms which can generate integers which are necessarily composite, yet not yield any information about what factors it has?
For example, $f(x):=x^2-1$ is not what I'm after, as you immediately know it has factors of $x-1$ and $x+1$.
Closer are Sierpinski numbers and primefree sequences, but my understanding is that both of them are fundamentally built on top of small, finite covering sets, such that it would be trivial to spit out an element of that covering set which would divide any of those numbers.
The only approach I know of that would work is to simply pick some arbitrarily large number at random and do a Miller-Rabin test or similar to confirm compositeness and then return it, so I'll go ahead and say that's not what I'm looking for either. It's hacky, and ideally the generating function would work for any input, not just spot-check and prune.