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Sequence of integers with divisibility condition

Source: APMO 2015 Problem 5

March 30, 2015
Sequencenumber theoryAPMO

Problem Statement

Determine all sequences a0,a1,a2,…a_0 , a_1 , a_2 , \ldots of positive integers with a0≥2015a_0 \ge 2015 such that for all integers n≥1n\ge 1: (i) an+2a_{n+2} is divisible by ana_n ; (ii) ∣sn+1−(n+1)an∣=1|s_{n+1} - (n + 1)a_n | = 1, where sn+1=an+1−an+an−1−⋯+(−1)n+1a0s_{n+1} = a_{n+1} - a_n + a_{n-1} - \cdots + (-1)^{n+1} a_0 .
Proposed by Pakawut Jiradilok and Warut Suksompong, Thailand
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