Problem

Source: 2019 China North MO

Tags: number theory



For positive intenger $n$, define $f(n)$: the smallest positive intenger that does not divide $n$. Consider sequence $(a_n): a_1=a_2=1, a_n=a_{f(n)}+1(n\geq3)$. For example, $a_3=a_2+1=2,a_4=a_3+1=3$. (a) Prove that there exists a positive intenger $C$, for any positive intenger $n$, $a_n\leq C$. (b) Are there positive intengers $M$ and $T$, satisfying that for any positive intenger $n\geq M$, $a_n=a_{n+T}$.