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a_n=\frac{n}{d}-d where n= greatest divisor of n such that d\leq \sqrt{n}

Source: 2023 Mathematics Regional Olympiad of Mexico West P1

October 21, 2024
number theorydivisor

Problem Statement

For every positive integer nn we take the greatest divisor dd of nn such that d≤nd\leq \sqrt{n} and we define an=nd−da_n=\frac{n}{d}-d. Prove that in the sequence a1,a2,a3,...a_1,a_2,a_3,..., any non negative integer kk its in the sequence infinitely many times.
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