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Let \[s(n)=\sigma(n)-n=\sum_{\substack{d\mid n\\ d<n}}d\] be the sum of proper divisors function.

If $A\subset \mathbb{N}$ has density $0$ then $s^{-1}(A)$ must also have density $0$.

A conjecture of Erdős, Granville, Pomerance and Spiro. It is possible for $s(A)$ to have positive density even if $A$ has zero density (for example taking $A$ to be the product of two distinct primes). Erdős [Er73b] proved that there are sets $A$ of positive density such that $s^{-1}(A)$ is empty.

Pollack [Po] has shown that this is true if $A$ is the set of primes. Troupe [Tr] has shown that this is true if $A$ is the set of integers with unusually many prime factors.