On the maximal order of numbers in the 'factorisatio numerorum' problem

Abstract

Let m(n) be the number of ordered factorizations of n into factors larger than 1; for example, m(12)=8 (12, 2.6, 6.2, 3.4, 4.3, 2.2.3, 2.3.2, 3.2.2). Denote L=log n and LL=loglog n and let r=1.728... be the solution of zeta(r)=2.

We prove that for every e>0 we have for large n the bound m(n) n^r/exp(cL^{1/r}/LL^{1/r}) holds for infinitely many n (c>0 is constant).

Keywords

• maximal
• order
• numbers
• &apos
• factorisatio
• numerorum&apos
• problem