IL-GILLIS 2018 Problem 5

The sequence $a_n$ is defined for any $n\geq 10$ by the following inductive rule:

• $a_{10}=5778$
• If $a_n=0$ then $a_{n+1}=0$ .
• If $a_n\neq0$ then $a_{n+1}$ is the number whose base- $(n+1)$ representation equals the base $n$ representation of the number $a_n -1$ . [/list] For example, $a_{11}=5\cdot11^3+7\cdot11^2+7\cdot11^1+7\cdot11^0=7586$ $a_{12}=5\cdot12^3+7\cdot12^2+7\cdot12^1+6\cdot12^0=9738$ [list=a]
• Does there exist $n\geq10$ for which $a_n=0$ ?
• Is $a_{1,000,000}=0$ ?
• Is $a_{100^{100^{100}}}=0$ ?