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