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a_{n+1}=\frac{a_n}{p}+p, where p is the greatest prime factor of a_n

Source: 2024 Mathematics Regional Olympiad of Mexico West P5

October 21, 2024
number theoryrecurrence relation

Problem Statement

Consider a sequence of positive integers a1,a2,a3,...a_1,a_2,a_3,... such that a1>1a_1>1 and an+1=anp+p,a_{n+1}=\frac{a_n}{p}+p, where pp is the greatest prime factor of ana_n. Prove that for any choice of a1a_1, the sequence a1,a2,a3,...a_1,a_2,a_3,... has an infinite terms that are equal between them.
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