Integer complexity: the integer defect

Abstract

Define $\parallel n\parallel$ to be the complexity of $n$, the smallest number of 1s needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $\parallel n\parallel \ge 3{log}_{3}n$ for all $n$, leading this author and Zelinsky to define the defect of $n$, $\delta \left(n\right)$, to be the difference $\parallel n\parallel -3{log}_{3}n$. Meanwhile, in the study of addition chains, it is common to consider $s\left(n\right)$, the number of small steps of $n$, defined as $\ell \left(n\right)-\lfloor {log}_{2}n\rfloor$, an integer quantity, where $\ell \left(n\right)$ is the length of the shortest addition chain for $n$. So here we analogously define $D\left(n\right)$, the integer defect of $n$, an integer version of $\delta \left(n\right)$ analogous to $s\left(n\right)$. Note that $D\left(n\right)$ is not the same as $\lceil \delta \left(n\right)\rceil$.

We show that $D\left(n\right)$ has additional meaning in terms of the defect well-ordering we considered in 2015, in that $D\left(n\right)$ indicates which powers of $\omega$ the quantity $\delta \left(n\right)$ lies between when one restricts to $n$ with $\parallel n\parallel$ lying in a specified congruence class modulo $3$. We also determine all numbers $n$ with $D\left(n\right)\le 1$, and use this to generalize a result of Rawsthorne (1989).