Integer Complexity and Well-Ordering

Abstract

Define $\|n\|$ to be the complexity of $n$, the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $\|n\| \ge 3\log_3 n$ for all $n$. Define the defect of $n$, denoted $\delta(n)$, to be $\|n\| - 3\log_3 n$. In this paper, we consider the set $\mathscr{D} := \{\delta(n): n \ge 1 \}$ of all defects. We show that as a subset of the real numbers, the set $\mathscr{D}$ is well-ordered, of order type $\omega^\omega$. More specifically, for $k\ge 1$ an integer, $\mathscr{D}\cap[0,k)$ has order type $\omega^k$. We also consider some other sets related to $\mathscr{D}$, and show that these too are well-ordered and have order type $\omega^\omega$.