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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 $i\le200$ and $n \le 10^{9}$.

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.

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    $\begingroup$ If you suspect the sequence hits infinitely many times a number, it seems like $n$ will be "random" modulo $i+1$ with respect to $a_i$. Say you have just hit $a$. If $i$ is big and we have to wait a bit to hit a number divisible by $i+1$, you will have: $a, r-a, a+1, r-a+1, a+2, r-a+2 \ldots$ where $r$ is a "big" random number. Since you have the code, I would suggest to test the sequence with random numbers (increasingly bigger) to better reproduce the 'definitive regime'. If explicit arithmetic properties of that n are necessary I guess it's a hard problem. $\endgroup$
    – Andrea Marino
    Commented Jul 7, 2021 at 20:16
  • $\begingroup$ Thanks for your comment and suggestion, I agree in general. $\endgroup$
    – Alkan
    Commented Jul 7, 2021 at 20:33

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