Question

Consider the inverse factorial function, f(n) = k where k! is the greatest factorial <= n. I’ve been told that the inverse factorial function is O(log n / log log n). Is it true? Or is it just a really really good approximation to the asymptotic growth? The methods I tried all give things very close to log(n)/log log(n) (either a small factor or a small term in the denominator) but not quite.

Answer 1

Remember that, when we’re using O(…), constant factors don’t matter, and any term that grows more slowly than another term can be dropped. ~ means “is proportional to.”

If k is large, then n = k! ~ k^k. So log n ~ k log k, or k ~ log n / log k or k ~ log n / log(log n / log k) = log n / (log log n - log log k). Because n >> k we can drop the term in the denominator, and we get k ~ log n / log log n so k = O(log n / log log n).