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a_{n+1} = a_n+rad(a_n), product of all distinct prime factors

Source: 2022 Saudi Arabia March Camp Test p1 BMO + EGMO TST

May 13, 2024
algebranumber theory

Problem Statement

By rad(x)rad(x) we denote the product of all distinct prime factors of a positive integer nn. Given a∈Na \in N, a sequence (an)(a_n) is defined by a0=aa_0 = a and an+1=an+rad(an)a_{n+1} = a_n+rad(a_n) for all n≥0n \ge 0. Prove that there exists an index nn for which anrad(an)=2022\frac{a_n}{rad(a_n)} = 2022
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