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

Let m(n) be the number of ordered factorizations of ngreater-or-equal, slanted1 in factors larger than 1. We prove that for every ε>0image holds for all integers n>n0, while, for a suitable constant c>0,image holds for infinitely many positive integers n, where ρ=1.72864… is the positive real solution to ζ(ρ)=2. We investigate also arithmetic properties of m(n) and the number of distinct values of m(n).