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

AT&T

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Hans Havermann

11 Sykes Ave.

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Robert G. Selcoe

16214 Madewood St.

Cypress, TX 77429

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Vladimir Shevelev

Department of Mathematics

Ben-Gurion University of the Negev

Beer-Sheva 84105

Israel

N. J. A. Sloane

The OEIS Foundation Inc.

11 South Adelaide Ave.

Highland Park, NJ 08904

USA

Reinhard Zumkeller

Isabellastrasse 13

D-80798 Munich

Germany

**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)
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.

(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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