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Claim. L. Troupe, Divisor sums representable as the sum of two squares, Proc. Amer. Math. Soc. 148 (2020), no. 10, 4189--4202, Theorem 1.2: the number $B_s(x)$ of $n\le x$ for which $s(n)$ is a sum of two squares satisfies $B_s(x)\asymp x/(\log x)^{1/2}$ (card). The paper presents this as a special case of the conjecture of Problem 955: the sums of two squares have density zero, and so does their preimage.

Covers. The instance $A$ = the sums of two squares; the general assertion stays open.

Depends on. No page of this wiki.

Acceptance. Refereed: Proceedings of the American Mathematical Society, a journal. The site's commentary credits the result, but the site labels the problem OPEN, so no curator acceptance is listed.