Integer complexity: Representing numbers of bounded defect

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‖≥3log3⁡n for all n. Based on this, this author and Zelinsky defined [4] the “defect” of n, δ(n):=‖n‖−3log3⁡n, and this author showed that the set of all defects is a well-ordered subset of the real numbers [1]. This was accomplished by showing that for a fixed real number s, there is a finite set S of polynomials called “low-defect polynomials” such that for any n with δ(n)