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Integer Subsets with High Volume and Low Perimeter

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Published/Copyright: October 2, 2012
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Integers
From the journal Integers Volume 12 Issue 5

Abstract.

We explore a variation of the isoperimetric problem in which integer subsets take the role of geometric figures. More specifically, we consider the sequence introduced by Miller, Morgan, Newkirk, Pedersen and Seferis (2011) and described in OEIS A186053. We provide the first exact formulas for including recursive relations via auxiliary functions as well as concise and satisfying representations and even quasi-explicit formulas. We also discuss some of the intricate fractal-like symmetry of the sequence and the development of algorithms for computing . We conclude with open questions for further research.

Keywords: Isoperimetric Problem; Integer Subsets; Triangle Numbers; Recurrence Relations
Received: 2011-07-14
Revised: 2012-02-06
Accepted: 2012-04-07
Published Online: 2012-10-02
Published in Print: 2012-10-01

© 2012 by Walter de Gruyter Berlin Boston

Articles in the same Issue

  1. Masthead
  2. Two-Color Babylon
  3. On Abelian and Additive Complexity in Infinite Words
  4. Approximations of Additive Squares in Infinite Words
  5. The Googol-th Bit of the Erdős–Borwein Constant
  6. Odd Repdigits to Small Bases Are Not Perfect
  7. Enumeration of the Degree Sequences of Line-Hamiltonian Multigraphs
  8. The Characteristic Sequence and p-Orderings of the Set of d-th Powers of Integers
  9. Van der Waerden's Theorem on Homothetic Copies of {1,1+s,1+s+t}
  10. Sum-Products Estimates with Several Sets and Applications
  11. The 2-Adic, Binary and Decimal Periods of 1/3k Approach Full Complexity for Increasing k
  12. A Proof of Catalan's Convolution Formula
  13. Explicit Constructions of Large Families of Generalized More Sums Than Differences Sets
  14. The Impossibility of Certain Types of Carmichael Numbers
  15. Integer Subsets with High Volume and Low Perimeter
  16. Mean-Value Theorems for Multiplicative Arithmetic Functions of Several Variables
  17. A Generalized Diagonal Wythoff Nim
  18. A Recursive Process Related to a Partizan Variation of Wythoff
  19. The (r1,...,rp)-Stirling Numbers of the Second Kind
  20. Factor Frequencies in Languages Invariant Under Symmetries Preserving Factor Frequencies
  21. On k-Lehmer Numbers
https://doi.org/10.1515/integers-2012-0017
Keywords for this article
Isoperimetric Problem; Integer Subsets; Triangle Numbers; Recurrence Relations

Articles in the same Issue

  1. Masthead
  2. Two-Color Babylon
  3. On Abelian and Additive Complexity in Infinite Words
  4. Approximations of Additive Squares in Infinite Words
  5. The Googol-th Bit of the Erdős–Borwein Constant
  6. Odd Repdigits to Small Bases Are Not Perfect
  7. Enumeration of the Degree Sequences of Line-Hamiltonian Multigraphs
  8. The Characteristic Sequence and p-Orderings of the Set of d-th Powers of Integers
  9. Van der Waerden's Theorem on Homothetic Copies of {1,1+s,1+s+t}
  10. Sum-Products Estimates with Several Sets and Applications
  11. The 2-Adic, Binary and Decimal Periods of 1/3k Approach Full Complexity for Increasing k
  12. A Proof of Catalan's Convolution Formula
  13. Explicit Constructions of Large Families of Generalized More Sums Than Differences Sets
  14. The Impossibility of Certain Types of Carmichael Numbers
  15. Integer Subsets with High Volume and Low Perimeter
  16. Mean-Value Theorems for Multiplicative Arithmetic Functions of Several Variables
  17. A Generalized Diagonal Wythoff Nim
  18. A Recursive Process Related to a Partizan Variation of Wythoff
  19. The (r1,...,rp)-Stirling Numbers of the Second Kind
  20. Factor Frequencies in Languages Invariant Under Symmetries Preserving Factor Frequencies
  21. On k-Lehmer Numbers
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