Refactoring factorials

SplittingN! intoN factors is trivial. It'sdefined as the product ofN integers, the integers 1 throughN.

Can you still splitN! intoN 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

Definet(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. Thet(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 ont(N) would be theNth root ofN! because that would correspond to splitting N! intoN factors of equal size. But the Nth root ofN! 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 largeN

t(N) /N 1/e - c₀ / log(N)

and the new paper proves

t(N) /N 1/e - c₀ / log(N) +O(1 / log^{1 + c}N).

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