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Numbers with Integer Complexity Close to the Lower Bound

1Department of Mathematics, University of Michigan, Ann Arbor, Michigan, USA

2Department of Mathematics and Statistics, Boston University, Boston, Massachusetts, USA

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

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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

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