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Numbers with Integer Complexity Close to the Lower Bound
-
Harry Altman
and
Joshua Zelinsky
Published/Copyright:
November 30, 2012
Abstract.
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.
Received: 2011-06-28
Revised: 2012-06-25
Accepted: 2012-07-24
Published Online: 2012-11-30
Published in Print: 2012-12-01
© 2012 by Walter de Gruyter Berlin Boston
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Articles in the same Issue
- Masthead
- Preface to the John Selfridge Memorial Issue
- Numbers with Integer Complexity Close to the Lower Bound
- On a Conjecture Regarding Balancing with Powers of Fibonacci Numbers
- Perfect Powers with Few Ternary Digits
- On Mullin's Second Sequence of Primes
- Log-Sine Evaluations of Mahler Measures, II
- On Odd Perfect Numbers and Even 3-Perfect Numbers
- Euler Pseudoprimes for Half of the Bases
- Odd Incongruent Restricted Disjoint Covering Systems
- Cubes in {0,1,...,n}3
- Sierpiński Numbers in Imaginary Quadratic Fields
- On a Partition Problem of Canfield and Wilf
- The 392 Problem
- Artin's Primitive Root Conjecture – A Survey
- Prime-Perfect Numbers
- Explicit Solutions of Certain Systems of Pell Equations
- The Search for Aurifeuillian-Like Factorizations
- Some Monoapparitic Fourth Order Linear Divisibility Sequences
- A Note from the Editors
Articles in the same Issue
- Masthead
- Preface to the John Selfridge Memorial Issue
- Numbers with Integer Complexity Close to the Lower Bound
- On a Conjecture Regarding Balancing with Powers of Fibonacci Numbers
- Perfect Powers with Few Ternary Digits
- On Mullin's Second Sequence of Primes
- Log-Sine Evaluations of Mahler Measures, II
- On Odd Perfect Numbers and Even 3-Perfect Numbers
- Euler Pseudoprimes for Half of the Bases
- Odd Incongruent Restricted Disjoint Covering Systems
- Cubes in {0,1,...,n}3
- Sierpiński Numbers in Imaginary Quadratic Fields
- On a Partition Problem of Canfield and Wilf
- The 392 Problem
- Artin's Primitive Root Conjecture – A Survey
- Prime-Perfect Numbers
- Explicit Solutions of Certain Systems of Pell Equations
- The Search for Aurifeuillian-Like Factorizations
- Some Monoapparitic Fourth Order Linear Divisibility Sequences
- A Note from the Editors
