Problems in $\mathsf{BPP}$ not known to be in $\mathsf P$? What problems are known to belong to $\mathsf{BPP}$ but not known to belong to $\mathsf P$?

More precisely, I am interested in independent problems, that is whose derandomizations are not known to be equivalent. For instance, it is known that derandomizing PIT and multivariate polynomial factorization are equivalent and I would count them as only one problem. 

The motivation of my question is that it is common to say that "there are few problems in $\mathsf{BPP}$ not known to be in $\mathsf{P}$", but I was not able to find a list of them. In particular, if I have to cite problems in this category, I usually cite the factorization of univariate polynomials over finite fields, or the factorization of multivariate polynomials. I suppose that there exist examples that are not related to polynomial factorization, for instance in other domains such as graph theory or formal language theory.

P.S.: I find curious that a similar question does not exist on this website yet. My apologies if I simply did not find it (or them)!
If you're asking for independent problems, how about:


  Find a prime in the interval $[N, 5N/4]$,
  Find two primes whose product is in the interval $[N, 9N/8]$,
  Find three primes whose product is in the interval $[N, 17N/16]$,
  Find four primes whose product is in the interval $[N, 33N/32]$,
  Find five primes whose product is in the interval $[N, 65N/64]$,
  $\ldots$. 


It's overwhelmingly likely that if you actually had a polynomial algorithm to solve the first of these, you would have a polynomial algorithm for all of them. But I don't see how to formally reduce any of these to any of the others. Of course, the problem


  Find a prime in the interval $[N, N+\log^{17} N ]$


solves all of these.