  Let $s(n)$ denote the sum of the proper divisors of the natural number $n$.
We show that the number of $n \leq x$ such that $s(n)$ is a sum of two squares
has order of magnitude $x/\sqrt{\log x}$, which agrees with the count of $n
\leq x$ which are a sum of two squares. Our result confirms a special case of a
conjecture of Erd{\H o}s, Granville, Pomerance and Spiro, who in a 1990 paper
asserted that if $\mathcal{A} \subset \mathbb{N}$ has asymptotic density zero
(e.g. if $\mathcal{A}$ is the set of $n \leq x$ which are a sum of two
squares), then $s^{-1}(\mathcal{A})$ also has asymptotic density zero.
