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Weakly Consecutive Sequences

Thomas Garrison, Chris Seiler, Andrew Knowles

Abstract

A weakly consecutive sequence (WCS) is a permutation $σ$ of $\{1, \ldots, k\}$ such that if an integer $d$ divides $σ(i)$, then $d$ also divides $σ(i \pm d)$ insofar as these are defined. The structure of weakly consecutive sequences is surprisingly rich, and it is difficult to find a formula for the number $N(k)$ of WCS's of length $k$. However, for a given $k$ we describe four starting sequences, to each of which we can apply three \emph{rules} or operations to generate new WCS's. We conjecture that any WCS can be constructed by applying these rules, which depend in an intricate way on the primality of $k$ and surrounding integers. We find bounds for $N(k)$ by analyzing these rules.

Weakly Consecutive Sequences

Abstract

A weakly consecutive sequence (WCS) is a permutation of such that if an integer divides , then also divides insofar as these are defined. The structure of weakly consecutive sequences is surprisingly rich, and it is difficult to find a formula for the number of WCS's of length . However, for a given we describe four starting sequences, to each of which we can apply three \emph{rules} or operations to generate new WCS's. We conjecture that any WCS can be constructed by applying these rules, which depend in an intricate way on the primality of and surrounding integers. We find bounds for by analyzing these rules.
Paper Structure (11 sections, 13 theorems, 49 equations, 3 tables)

This paper contains 11 sections, 13 theorems, 49 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Starting sequences and rules
  3. General results
  4. Generating weakly consecutive sequences
  5. Prime Power Swaps
  6. Notation:
  7. Lower bounds on #I(k) and N(k)
  8. Upper bounds on liminf #I(k)
  9. Conjectures and future work
  10. Deferred proofs
  11. Acknowledgements

Key Result

Theorem 1.2

Fix an $n \in \mathbb{Z}^+$. There exists a $k \leq 2^{n^{2 \lg(\lg (n)) (1 + o(1))}}$ with $N(k) \geq n$.

Theorems & Definitions (39)

  • Definition 1.1
  • Theorem 1.2: see Corollary \ref{['cor:nk']} below
  • Definition 2.1
  • Lemma 2.2: Division slice characterization of weakly consecutive sequence
  • proof
  • Remark
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 29 more