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.