  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].
