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

Let m(n)m(n) be the number of ordered factorizations of n⩾1n⩾1 in factors larger than 1. We prove that for every ε>0ε>0m(n)n0n>n0, while, for a suitable constant c>0c>0,m(n)>nρexp(c(logn/loglogn)1/ρ) holds for infinitely many positive integers n  , where ρ=1.72864…ρ=1.72864… is the positive real solution to ζ(ρ)=2ζ(ρ)=2. We investigate also arithmetic properties of m(n)m(n) and the number of distinct values of m(n)m(n).