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A bijection on N

Source: Tournament of Towns, Fall 2002, Senior A Level, P6

May 17, 2014
number theorygreatest common divisornumber theory proposed

Problem Statement

Define a sequence {an}n≥1\{a_n\}_{n\ge 1} such that a1=1,a2=2a_1=1,a_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)≠1\text{gcd}(m,a_n)\neq 1. Show all positive integers occur in the sequence.
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