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Omni-MATH / Given an integer m, define the sequence {a_(n)} as follows:

Problem

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problem

Given an integer m m, define the sequence {an} \left\{a_{n}\right\} as follows: \[ a_{1}\equal{}\frac{m}{2},\ a_{n\plus{}1}\equal{}a_{n}\left\lceil a_{n}\right\rceil,\textnormal{ if }n\geq 1\] Find all values of m m for which a2007 a_{2007} is the first integer appearing in the sequence. Note: For a real number x x, $ \left\lceil x\right\rceil$ is defined as the smallest integer greater or equal to x x. For example, $ \left\lceil\pi\right\rceil\equal{}4$, $ \left\lceil 2007\right\rceil\equal{}2007$.
Plain-text mathematical notation (without MathML)
Given an integer m, define the sequence {a_(n)} as follows:
\[ a_{1}\equal{}\frac{m}{2},\ a_{n\plus{}1}\equal{}a_{n}\left\lceil a_{n}\right\rceil,\textnormal{ if }n\geq 1\]
Find all values of m for which a₂₀₀₇ is the first integer appearing in the sequence.

Note: For a real number x, $ \left\lceil x\right\rceil$ is defined as the smallest integer greater or equal to x. For example, $ \left\lceil\pi\right\rceil\equal{}4$, $ \left\lceil 2007\right\rceil\equal{}2007$.
Original LaTeX notation
Given an integer $ m$, define the sequence $ \left\{a_{n}\right\}$ as follows:
\[ a_{1}\equal{}\frac{m}{2},\ a_{n\plus{}1}\equal{}a_{n}\left\lceil a_{n}\right\rceil,\textnormal{ if }n\geq 1\]
Find all values of $ m$ for which $ a_{2007}$ is the first integer appearing in the sequence.

Note: For a real number $ x$, $ \left\lceil x\right\rceil$ is defined as the smallest integer greater or equal to $ x$. For example, $ \left\lceil\pi\right\rceil\equal{}4$, $ \left\lceil 2007\right\rceil\equal{}2007$.

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Source and history

task_id
8129941b-ad9f-5f4a-8a89-822a94336d59
task_key
test--8129941b-ad9f-5f4a-8a89-822a94336d59
task_revision_id
1
config / split
/ test

Official source

initial import