On a Conjecture about Ron Graham's Sequence

TL;DR

This paper explores Ron Graham's Sequence, providing properties, an algorithm for computation, and proving a long-standing conjecture about its upper bound.

Contribution

It introduces an algorithm for computing the sequence efficiently and proves a 22-year-old conjecture regarding its upper bound.

Findings

The sequence is a bijection from non-negative integers to non-prime integers.

An algorithm for pseudo-polynomial time computation is presented.

A conjecture about the sequence's upper bound is proven.

Abstract

Ron Graham's Sequence is a surprising bijection from non-negative integers to non-negative, non-prime integers that was introduced by Ron Graham in the June 1986 "Problems" column of $\textit{Mathematics Magazine}$, and which later appeared in Problem A2 of the 2013 William Lowell Putnam Mathematical Competition. We describe some properties of this function, give an algorithm for computing its values in pseudo-polynomial time, and prove a 22 year-old conjecture about an upper bound for the function.