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

Let ω(n)ω(n) (resp. Ω(n)Ω(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 ω(n)ω(n) is log⁡log⁡nlog⁡log⁡n, and the same is true of Ω(n)Ω(n); roughly speaking, a typical natural number n   has about log⁡log⁡nlog⁡log⁡n prime factors. We prove a similar result for ω(s(n))ω(s(n)), where s(n)s(n) denotes the sum of the proper divisors of n  : For all n≤xn≤x not belonging to a set of size o(x)o(x),|ω(s(n))−log⁡log⁡s(n)|<ϵlog⁡log⁡s(n),|ω(s(n))−log⁡log⁡s(n)|<ϵlog⁡log⁡s(n), and the same is true for Ω(s(n))Ω(s(n)).