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Interesting recurrence

Source: Peru Cono Sur TST 2020 P6

September 11, 2023
number theory

Problem Statement

Let a1,a2,a3,…a_1, a_2, a_3, \ldots a sequence of positive integers that satisfy the following conditions: a1=1,an+1=an+a⌊n⌋,∀n≥1a_1=1, a_{n+1}=a_n+a_{\lfloor \sqrt{n} \rfloor}, \forall n\ge 1 Prove that for every positive integer kk there exists a term aia_i that is divisible by kk
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