On a Generalisation of a Function of Ron Graham's

Ron Graham introduced a function, $g(n)$, on the non-negative integers, in the 1986 Issue $3$ Problems column of $\mathit{\text{Mathematical}}\mathit{\text{Magazine}}$: For each non-negative integer $n$, $g(n)$ is the least integer $s$ so that the integers $n+1,n+2,\dots ,s-1,s$ 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 $m\ge 2$, 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.