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Sequence problem

Source: Azerbaijan Math Olympiad Training

December 14, 2019
number theoryTST

Problem Statement

Define a sequence ann≥1{{a_n}}_{n\ge1} such that a1=1a_1=1 , a2=2a_2=2 and an+1a_{n+1} is the smallest positive integer mm such that mm hasn't yet occurred in the sequence and also gcd(m,an)≠1gcd(m,a_n)\neq{1}. Show that all positive integers occur in the sequence.
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