If I understand your problem correctly, the tokens (lanterns) can be placed every $x$ blocks (starting from $0$) if and only if $x>2$ is a divisor of $n-1$.

For example, if the arrays has $n=31$ elements the valid values of $x$ are $3,5,6,10,15,$ and $30$.

If I understand your problem correctly, the tokens (lanterns) can be placed every $x$ blocks (starting from $0$) if and only if $x>2$ is a divisor of $n-1$.

For example, if the array has $n=31$ elements the valid values of $x$ are $3,5,6,10,15,$ and $30$.