The Yellowstone Permutation

TL;DR

This paper introduces the Yellowstone permutation, a sequence of positive integers with a unique rule, and explores its complex graph structure resembling geyser eruptions, revealing intricate patterns and raising open questions about its properties.

Contribution

The paper defines a novel permutation sequence based on specific divisibility and coprimality rules and analyzes its intricate graphical behavior and underlying patterns.

Findings

Sequence is a permutation of positive integers.

Graph exhibits alternating even-odd runs with spikes.

Points align along infinitely many curves.

Abstract

Define a sequence of positive integers by the rule that a(n) = n for 1 <= n <= 3, and for n >= 4, a(n) is the smallest number not already in the sequence which has a common factor with a(n-2) and is relatively prime to a(n-1). We show that this is a permutation of the positive integers. The remarkable graph of this sequence consists of runs of alternating even and odd numbers, interrupted by small downward spikes followed by large upward spikes, suggesting the eruption of geysers in Yellowstone National Park. On a larger scale the points appear to lie on infinitely many distinct curves. There are several unanswered questions concerning the locations of these spikes and the equations for these curves.