Volume 12 (2012)
Volume 11 (2011)
Volume 10 (2010)
Most Downloaded Articles
- On the Number of Carries Occurring in an Addition Mod by Flori, Jean-Pierre and Randriam, Hugues
- An Explicit Bound for Aliquot Cycles of Repdigits by Broughan, Kevin A.
- Number of Permutations with Prescribed Up-Down Structure as a Function of Two Variables by Shevelev, Vladimir
- A -Expansion Associated to Sturmian Sequences by Nakano, Fumihiko and Sadahiro, Taizo
- Avoiding Type or Patterns in a Partition of a Set by Mansour, Toufik and Shattuck, Mark
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: . Volume 12, Issue 6, Pages 1093–1125, ISSN (Online) 1867-0652, ISSN (Print) 1867-0652, DOI: 10.1515/integers-2012-0031, November 2012
- Published Online:
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.