On a Generalisation of a Function of Ron Graham's
Sarosh Adenwalla
TL;DR
This work generalises Graham’s function $g(n)$ to $g_m(n)$, the least endpoint of $m$-product sequences starting at $n$, and establishes a comprehensive picture across prime and composite values of $m$. It proves the conjectured $g_2(n)=2n$ characterization in one direction (the 'only if' part: $g_2(n)=2n$ implies $n$ is prime and $n>3$, or $n\in\{0,6\}$), and extends the analysis to general $m$ by deriving upper bounds, bijectivity for prime $m$, and non-injectivity/non-surjectivity for composite $m$. The paper also introduces and studies the length parameters $T_m(n)$ and $T'_m(n)$ of corresponding sequences, providing structural results such as $T'_m(n)\le T_m(n)$ and explicit formulas when $g_m(n)=n$. Collectively, these results clarify the arithmetic structure of $g_m(n)$, reveal how prime versus composite $m$ governs injectivity/surjectivity, and lay groundwork for further exploration of $m$-th power generalisations and their combinatorial properties.
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.