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Nice Algebra Problem

Source: Saint-Peterburg Math Olympiad 2017. Class 10. Problem 2

January 30, 2018
algebra

Problem Statement

(an)(a_{n}) is sequence with positive integer. a1>10a_{1}>10 an=an−1+GCD(n,an−1) a_{n}=a_{n-1}+GCD(n,a_{n-1}), n>1 For some i ai=2ia_{i}=2i. Prove that these numbers are infinite in this sequence.
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