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sequence with greatest prime factor

Source: IMOC 2021 N3

August 12, 2021
number theorySequences

Problem Statement

Define the function f:N>1→N>1f:\mathbb N_{>1}\to\mathbb N_{>1} such that f(x)f(x) is the greatest prime factor of xx. A sequence of positive integers {an}\{a_n\} satisfies a1=M>1a_1=M>1 and an+1={an−f(an)if an is composite.an+kotherwise.a_{n+1}=\begin{cases}a_n-f(a_n)&\text{if }a_n\text{ is composite.}\\a_n+k&\text{otherwise.}\end{cases} Show that for any positive integers M,kM,k, the sequence {an}\{a_n\} is bounded.
(TAN768092100853)
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