MathDB

5

Part of 2022 Switzerland - Final Round

Problems(1)

a_{n+1} = a_n + \delta_(a_n) where \delta_(a) =second largest divisor of a

Source: Switzerland - 2022 Swiss Final Round p5

11/17/2022
For an integer a≥2a \ge 2, denote by δ(a)\delta_(a) the second largest divisor of aa. Let (an)n≥1(a_n)_{n\ge 1} be a sequence of integers such that a1≥2a_1 \ge 2 and an+1=an+δ(an)a_{n+1} = a_n + \delta_(a_n) for all n≥1n \ge 1. Prove that there exists a positive integer kk such that aka_k is divisible by 320223^{2022}.
number theoryrecurrence relationdivisor