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Omni-MATH / Determine all sequences a₀,a₁,a₂,… of positive integers with a₀≥2015 such that for all integers n≥1 : (i) a_(n+2) is divisible by a_(n); (ii) |s_(n+1)−(n+1)a_(n)|=1, where …

Problem

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problem

Determine all sequences a0,a1,a2,…a_{0}, a_{1}, a_{2}, \ldots of positive integers with a0≥2015a_{0} \geq 2015 such that for all integers n≥1n \geq 1 : (i) an+2a_{n+2} is divisible by ana_{n}; (ii) |sn+1−(n+1)an|=1\left|s_{n+1}-(n+1) a_{n}\right|=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}.
Plain-text mathematical notation (without MathML)
Determine all sequences a₀,a₁,a₂,… of positive integers with a₀≥2015 such that for all integers n≥1 : (i) a_(n+2) is divisible by a_(n); (ii) |s_(n+1)−(n+1)a_(n)|=1, where s_(n+1)=a_(n+1)−a_(n)+a_(n−1)−⋯+(−1)^(n+1)a₀.
Original LaTeX notation
Determine all sequences $a_{0}, a_{1}, a_{2}, \ldots$ of positive integers with $a_{0} \geq 2015$ such that for all integers $n \geq 1$ : (i) $a_{n+2}$ is divisible by $a_{n}$; (ii) $\left|s_{n+1}-(n+1) a_{n}\right|=1$, where $s_{n+1}=a_{n+1}-a_{n}+a_{n-1}-\cdots+(-1)^{n+1} a_{0}$.

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Source and history

task_id
14ad5c9b-6dda-53c0-83c0-62a911fb1fc5
task_key
test--14ad5c9b-6dda-53c0-83c0-62a911fb1fc5
task_revision_id
1
config / split
/ test

Official source

initial import