Let$$s(n)=\sigma (n)-n=\sum _{\begin{array}{c}d\mid n\\ d<n\end{array}}d$$be the sum of proper divisors function.If $A\subset \mathbb{N}$ has density $0$ then $s^{-1}(A)$ must also have density $0$.

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

What you wrote is (as far as I can tell) **not known in full generality**. It’s a well-known open conjecture about the aliquot sum / sum-of-proper-divisors function.

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.