Divisor sums representable as the sum of two squares.

Let s(n) denote the sum of the proper divisors of the natural number n. We show that the number of n ≤ x such that s(n) is a sum of two squares has order of magnitude x/√log x, which agrees with the count of n ≤ 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 has asymptotic density zero (e.g., if A is the set of n ≤ x which are a sum of two squares), then s⁻¹(A) also has asymptotic density zero. [ABSTRACT FROM AUTHOR]