This question is related to a family of sequences. I have a simple definition as below and I have a question based on my limited observations for $i\le100$ and $n \le 10^{8}$.

Definition. $a_{i}(1) = 1$. For $n>1$, $a_i(n) = a_i(n-1)/(i+1)$ if $a_i(n-1)$ is divisible by $i+1$, otherwise $a_i(n) = n - a_i(n-1)$.

For example, $a_{1}(n)$ is https://oeis.org/A345877 and $a_{2}(n)$ is https://oeis.org/A345886.

Question. Does $a_{i}(n)$ hit every positive integers infinitely many times for all $i\ge1$?

I will be grateful for any suggestion or reference about solution of above question.
Thanks.