Erdős Problem #955

declared status 'open'. Formalized: no. 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 $0$. Current best: Pollack, Pomerance, and Thompson [PPT18] prove that if $\epsilon(x)=o(1)$ and $A\subset \mathbb{N}$ has size at most $x^{1/2+\epsilon(x)}$ then$$\#\{ n\leq x: s(n)\in A\} =o(x)$$as $x\to \infty$. It follows that (using $s(n)\ll n\log\log n$) if $A$ grows like $\lvert A\cap [1,x]\rvert\leq x^{1/2+o(1)}$ then $s^{-1}(A)$ has density $0$. Prize: no. Tags: number theory.

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