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 wellordering 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).
