DE-BUNDESWETTBEWERB 2022 Problem 4

For each positive integer $k$ let $a_k$ be the largest divisor of $k$ which is not divisible by $3$ . Let $s_n=a_1+a_2+\dots+a_n$ . Show that: (a) The number $s_n$ is divisible by $3$ iff the number of ones in the ternary expansion of $n$ is divisible by $3$ . (b) There are infinitely many $n$ for which $s_n$ is divisible by $3^3$ .