Rational numbers with odd greedy expansion of fixed length

Abstract

Given a positive rational number $n/d$ with $d$ odd, its odd greedy expansion starts with the largest odd denominator unit fraction at most $n/d$, adds the largest odd denominator unit fraction so the sum is at most $n/d$, and continues as long as the sum is less than $n/d$. It is an open question whether this expansion always has finitely many terms. Given a fixed positive integer $n$, we find all reduced fractions with numerator $n$ whose odd greedy expansion has length $2$. Given $m-1$ odd positive integers, we find all rational numbers whose odd greedy expansion has length $m$ and begins with these numbers as denominators. Given $m-2$ compatible odd positive integers, we find an infinite family of rational numbers whose odd greedy expansion has length $m$ and begins with these numbers as denominators.