Splitting N! into N factors is trivial. It’s defined as the product of N integers, the integers 1 through N.
Can you still split N! into N factors if you put a minimum size requirement on the factors?
You can’t split 3! into three factors unless one of the factors is 1, but you can split 4! into factors ≥ 2:
4! = 24 = 2 × 2 × 2 × 3
Define t(N) as the largest number such that N! can be expressed as the product of integers greater than or equal to t(N). So based on the discussion above, t(N) ≥ 1 for all N, t(3) = 1 and t(4) = 2. The t(N) are given in OEIS sequence A034258.
Last Thursday an article came out proving an asymptotic estimate for t(N). You can find a preprint on arXiv.
An upper bound on t(N) would be the Nth root of N! because that would correspond to splitting N! into N factors of equal size. But the Nth root of N! will not be an integer for N > 1, so the bound would not be tight.
If we apply Stirling’s approximation for factorial we get
t(N) / N ≤ 1/e + O(log(N) / N).
It was conjectured that for large N
t(N) / N ∼ 1/e − c0 / log(N)
and the new paper proves
t(N) / N ∼ 1/e − c0 / log(N) + O(1 / log1 + c N).
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