On the number of prime factors of values of the sum-of-proper-divisors function

Abstract

Let $\omega(n)$ (resp. $\Omega(n)$) denote the number of prime divisors (resp. with multiplicity) of a natural number $n$. In 1917, Hardy and Ramanujan proved that the normal order of $\omega(n)$ is $\log\log n$, and the same is true of $\Omega(n)$; roughly speaking, a typical natural number $n$ has about $\log\log n$ prime factors. We prove a similar result for $\omega(s(n))$, where $s(n)$ denotes the sum of the proper divisors of $n$: For any $\epsilon > 0$ and all $n \leq x$ not belonging to a set of size $o(x)$, \[ |\omega(s(n)) - \log\log s(n)| < \epsilon \log\log s(n) \] and the same is true for $\Omega(s(n))$.