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Omni-MATH / Let S={p₁p₂⋯p_(n)∣p₁,p₂,…,p_(n) are distinct primes and p₁,…,p_(n)<30}. Assume 1 is in S. Let a₁ be an element of S. We define, for all positive integers n : …

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problem

Let S={p1p2⋯pn∣p1,p2,…,pnS=\left\{p_{1} p_{2} \cdots p_{n} \mid p_{1}, p_{2}, \ldots, p_{n}\right. are distinct primes and p1,…,pn<30}\left.p_{1}, \ldots, p_{n}<30\right\}. Assume 1 is in SS. Let a1a_{1} be an element of SS. We define, for all positive integers nn : $$ \begin{gathered} a_{n+1}=a_{n} /(n+1) \quad \text { if } a_{n} \text { is divisible by } n+1 \\ a_{n+1}=(n+2) a_{n} \quad \text { if } a_{n} \text { is not divisible by } n+1 \end{gathered} $$ How many distinct possible values of a1a_{1} are there such that aj=a1a_{j}=a_{1} for infinitely many jj 's?
Plain-text mathematical notation (without MathML)
Let S={p₁p₂⋯p_(n)∣p₁,p₂,…,p_(n) are distinct primes and p₁,…,p_(n)<30}. Assume 1 is in S. Let a₁ be an element of S. We define, for all positive integers n : $$ \begin{gathered} a_{n+1}=a_{n} /(n+1) \quad \text { if } a_{n} \text { is divisible by } n+1 \\ a_{n+1}=(n+2) a_{n} \quad \text { if } a_{n} \text { is not divisible by } n+1 \end{gathered} $$ How many distinct possible values of a₁ are there such that a_(j)=a₁ for infinitely many j 's?
Original LaTeX notation
Let $S=\left\{p_{1} p_{2} \cdots p_{n} \mid p_{1}, p_{2}, \ldots, p_{n}\right.$ are distinct primes and $\left.p_{1}, \ldots, p_{n}<30\right\}$. Assume 1 is in $S$. Let $a_{1}$ be an element of $S$. We define, for all positive integers $n$ : $$ \begin{gathered} a_{n+1}=a_{n} /(n+1) \quad \text { if } a_{n} \text { is divisible by } n+1 \\ a_{n+1}=(n+2) a_{n} \quad \text { if } a_{n} \text { is not divisible by } n+1 \end{gathered} $$ How many distinct possible values of $a_{1}$ are there such that $a_{j}=a_{1}$ for infinitely many $j$ 's?

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Source and history

task_id
52f92730-d7b3-58a9-b3be-d4d71d8145b5
task_key
test--52f92730-d7b3-58a9-b3be-d4d71d8145b5
task_revision_id
1
config / split
/ test

Official source

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