Problem 1656

In order to solve this, we will give a complete classification of $n$-sequences. Let $k=\lfloor n / 2\rfloor$. We will say that an $n$-sequence is large if $a_{i}>k$ for some $i$, and small if no such $i$ exists. For now we will assume that $\left(a_{i}\right)$ is not the identity sequence (in other words, that $a_{i} \neq i$ for some $i$ ). Lemma 1. If $a_{r}=a_{s}$ and $r, s < n$, and let $d$ be the minimum positive integer such that $a_{r+d}=a_{r}$. Then 1. The subsequence $\left(a_{r}, a_{r+1}, \ldots, a_{n}\right)$ is periodic with minimal period $d$. That is, for $u, v \geq r$, $a_{u}=a_{v}$ if and only if $d \mid u-v$. 2. If $r=0$ there is nothing to prove. Otherwise $a_{0}=a_{2 a_{0}}$ so $2 a_{0}=0$. Then we have $a_{a_{i}}=a_{i}$ for all $i$, so $a_{i}=i$ for $i \leq k$. Lemma 2. If $\left(a_{i}\right)$ is a small $n$-sequence, then $a_{i} \leq k$ for all $i$. Proof. We show that $a_{i} \leq k$ for all $i$ by induction. 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