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Omni-MATH / For each integer a₀>1, define the sequence a₀,a₁,a₂,… for n≥0 as …

Problem

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problem

For each integer a0>1a_0 > 1, define the sequence a0,a1,a2,…a_0, a_1, a_2, \ldots for n≥0n \geq 0 as $$a_{n+1} = \begin{cases} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\ a_n + 3 & \text{otherwise.} \end{cases} $$ Determine all values of a0a_0 such that there exists a number AA such that an=Aa_n = A for infinitely many values of nn. [i]
Plain-text mathematical notation (without MathML)
For each integer a₀>1, define the sequence a₀,a₁,a₂,… for n≥0 as
$$a_{n+1} = 
\begin{cases}
\sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\
a_n + 3 & \text{otherwise.}
\end{cases}
$$
Determine all values of a₀ such that there exists a number A such that a_(n)=A for infinitely many values of n.

[i]
Original LaTeX notation
For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as
$$a_{n+1} = 
\begin{cases}
\sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\
a_n + 3 & \text{otherwise.}
\end{cases}
$$
Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitely many values of $n$.

[i]

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Source and history

task_id
2f11bda5-4349-5b41-b605-c29628d16eda
task_key
test--2f11bda5-4349-5b41-b605-c29628d16eda
task_revision_id
1
config / split
/ test

Official source

initial import