On the maximal order of numbers in the “factorisatio numerorum” problem

Let m(n) be the number of ordered factorizations of n ≥ 1 in factors larger than 1. We prove that for every ε > 0 m(n) < nρ exp ( (log n)1/ρ/(log log n)1+ε ) holds for all integers n > n0, while, for a constant c > 0, m(n) > nρ exp ( c(log n/ log log n)1/ρ ) holds for infinitely many positive integers n, where ρ = 1.72864 . . . is the real solution to ζ(ρ) = 2. We investigate also arithmetic properties of m(n) and the number of distinct values of m(n).