  Let $s(n)=\sum_{d\mid n,~d<n} d$ denote the sum of the proper divisors of
$n$. The second-named author proved that $\omega(s(n))$ has normal order
$\log\log{n}$, the analogue for $s$-values of a classical result of Hardy and
Ramanujan. We establish the corresponding Erd\H{o}s--Kac theorem:
$\omega(s(n))$ is asymptotically normally distributed with mean and variance
$\log\log{n}$. The same method applies with $s(n)$ replaced by any of several
other unconventional arithmetic functions, such as $\beta(n):=\sum_{p\mid n}
p$, $n-\phi(n)$, and $n+\tau(n)$ ($\tau$ being the divisor function).
