On a Generalisation of a Function of Ron Graham's

Abstract: Ron Graham introduced a function, $g(n)$, on the non-negative integers, in the 1986 Issue $3$ Problems column of \textit{Mathematical Magazine}: For each non-negative integer $n$, $g(n)$ is the least integer $s$ so that the integers $n + 1, n + 2, \ldots , s-1, s$ contain a subset of integers, the product of whose members with $n$ is a square. Recently, many results about $g(n)$ were proved in [Kagey and Rajesh, ArXiv:2410.04728, 2024] and they conjectured a characterization of which $n$ satisfied $g(n)=2n$. For $m\geq 2$, they also introduced generalizations of $g(n)$ to $m$-th powers to explore. In this paper, we prove their conjecture and provide some results about these generalisations.