The yellowstone permutation

David L. Applegate, Hans Havermann, Robert G. Selcoe, Vladimir Shevelev, N. J.A. Sloane, Reinhard Zumkeller

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


Define a sequence of positive integers by the rule that a(n) = n for 1 £ n £ 3, and for n ³ 4, a(n) is the smallest number not already in the sequence which has a common factor with a(n − 2) but 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.

Original languageEnglish
JournalJournal of Integer Sequences
Issue number6
StatePublished - 1 Jan 2015


  • EKG sequence
  • Number sequence
  • Permutation of natural numbers

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics


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