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Numbers with Integer Complexity Close to the Lower Bound

Harry Altman / Joshua Zelinsky
Published Online: 2012-11-30 | DOI: https://doi.org/10.1515/integers-2012-0031


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.

Keywords: Integer Complexity; Integer Sequence; Arithmetic; Arithmetic Operations

About the article

Received: 2011-06-28

Revised: 2012-06-25

Accepted: 2012-07-24

Published Online: 2012-11-30

Published in Print: 2012-12-01

Citation Information: Integers, Volume 12, Issue 6, Pages 1093–1125, ISSN (Online) 1867-0652, ISSN (Print) 1867-0652, DOI: https://doi.org/10.1515/integers-2012-0031.

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© 2012 by Walter de Gruyter Berlin Boston.Get Permission

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