Journal of Integer Sequences, Vol. 18 (2015), Article 15.6.7

The Yellowstone Permutation

David L. Applegate
One AT&T Way
Bedminster, NJ 07921

Hans Havermann
11 Sykes Ave.
Weston, ON M9N 1C8

Robert G. Selcoe
16214 Madewood St.
Cypress, TX 77429

Vladimir Shevelev
Department of Mathematics
Ben-Gurion University of the Negev
Beer-Sheva 84105

N. J. A. Sloane
The OEIS Foundation Inc.
11 South Adelaide Ave.
Highland Park, NJ 08904

Reinhard Zumkeller
Isabellastrasse 13
D-80798 Munich


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) 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.

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(Concerned with sequences A006368 A064413 A098548 A098550 A249167 A249943 A251237 A251411 A251412 A251413 A251542 A251543 A251544 A251545 A251546 A251547 A251554 A251555 A251556 A251557 A251558 A251559 A251604 A251621 A251756 A252837 A252838 A252865 A252867 A252868 A253048 A253049.)

Received March 7 2015; revised version received June 10 2015. Published in Journal of Integer Sequences, June 13 2015.

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