Integer complexity: The integer defect

TL;DR

This paper introduces the integer defect $D(n)$ as an integer analogue of the defect $\,\, ext{delta}(n)$ in integer complexity, explores its properties within well-ordering, and characterizes numbers with $D(n)\,\, ext{less than or equal to}\, 1$.

Contribution

The paper defines the integer defect $D(n)$, analyzes its relation to the well-ordering of defect values, and characterizes all numbers with $D(n)\,\, ext{≤ 1}$, extending previous results.

Findings

$D(n)$ indicates the position of $\, ext{delta}(n)$ in the well-ordering.

All numbers with $D(n) \, ext{≤ 1}$ are characterized.

Generalization of Rawsthorne's result is achieved.

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 $n$, leading this author and Zelinsky to define the defect of $n$, $\delta(n)$, to be the difference $\|n\|-3\log_3 n$. Meanwhile, in the study of addition chains, it is common to consider $s(n)$, the number of small steps of $n$, defined as $\ell(n)-\lfloor\log_2 n\rfloor$, an integer quantity. So here we analogously define $D(n)$, the integer defect of $n$, an integer version of $\delta(n)$ analogous to $s(n)$. Note that $D(n)$ is not the same as $\lceil \delta(n) \rceil$. We show that $D(n)$ has additional meaning in terms of the defect well-ordering considered in [3], in that $D(n)$ indicates which powers of $\omega$ the quantity $\delta(n)$ lies between when one restricts to $n$ with $\|n\|$ lying in a specified congruence class modulo $3$. We also determine all numbers $n$ with $D(n)\le 1$, and use this to generalize a result of Rawsthorne [18].