Find all such sequences $(x_1, x_2, x_3, ..., x_{63})$ consisting of different positive integers that for $n=1,2,3,...,62$ the number $x_n$ is a divisor of $x_{n+1}+1$ and $x_{63}$ is a divisor of $x_1+1$.
Suppose we let $x_{63}=k$. Then we know $x_1 = k+62$.
Our condition is that the last term is a factor of the first term plus 1. Another way of saying that is that
$\frac{k+62+1}{k}= \frac{k+63}{k}$ is an integer. So we have to find all the integer solutions for the equation
A better way of doing it is noticing that we can rewrite the equation as: $k(n-1)=63$
So we know that there are 6 such sequences as there are 6 factors of 63.
And here is the problem: are the sequences of 62+k, 62+k-1, 62+k-2,... the only ones which can fulfill the requirements? I mean, look: $\{1,3,14,27,14039,...,some\_positive\_integer\}$ also does while not being either descending nor having any particular "step". So are the ones I found really the only ones?