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All numbers occur exactly once

Source: St. Petersburg Olympiad 2015, 2nd Round, Grade 9, also known as Yellowstone Permutation

September 24, 2016
number theoryrelatively prime

Problem Statement

A sequence of integers is defined as follows: a1=1,a2=2,a3=3a_1=1,a_2=2,a_3=3 and for n>3n>3, an=The smallest integer not occurring earlier, which is relatively prime to an−1 but not relatively prime to an−2.a_n=\textsf{The smallest integer not occurring earlier, which is relatively prime to }a_{n-1}\textsf{ but not relatively prime to }a_{n-2}.Prove that every natural number occurs exactly once in this sequence.
M. Ivanov
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