## Abstract.

Define 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
for all *n*. Define the *defect* of *n*, denoted by , to be
; in this paper we present a method for classifying all *n*
with for a given *r*. From this, we derive several
consequences. We prove that for with *m* and
*k* not both zero, and present a method that can, with more computation,
potentially prove the same for larger *m*. Furthermore, defining to be
the number of *n* with and , we prove that
, allowing us to conclude that
the values of can be arbitrarily large.

## Comments (0)