"How many $1$'s are needed?" revisited

TL;DR

This paper introduces a rigorous and efficient method for computing the complexity of natural numbers, extends related sequences, and discusses historical and collaborative aspects of the research.

Contribution

It provides the first rigorous proof of the correctness of a program for computing number complexity and extends known terms of related sequences.

Findings

Proved the correctness of a program for number complexity

Extended the sequence A005520 with new terms

Presented a fast method for complexity computation

Abstract

We present a rigorous and relatively fast method for the computation of the "complexity" of a natural number (sequence A005245), and answer some "old and new" questions related to the question in the title of this note. We also extend the known terms of the related sequence A005520. We put this paper in the arXiv only for possible reference. This note was written in 2008. It is exactly the copy we have sent to Martin N. Fuller in February 11, 2009 proposing to him to join us as co-author. The ensuing collaboration resulted in a better program and many results about the complexity. We have waited four years in the hope of contacting him again. In the mean time Iraids et al. published arXiv:1203.6462 containing some of our results here. Our program is similar to the one posted in the OEIS by Martin N. Fuller, but its correctness is proved here for the first time.