Amicable numbers

Abstract

Let $s(n):={\sum }_{d\mid n,d<n}d$ denote the sum of the proper divisors of $n$. Distinct positive integers $n,m$ form an amicable pair if $s(n)=m$ and $s(m)=n$. Let $A(x)$ count the positive integers not exceeding $x$ that belong to an amicable pair. We report on a proof, found by ChatGPT Astra, that $$A(x)\le x\exp \{-(1/2+o(1))\log x{\log }_{3}x/{\log }_{2}x\}$$ as $x\to \infty$. Here ${\log }_k$ denotes the $k$th iterate of the natural logarithm.

11 pages; expository article on AI-generated work