# Terms of a certain recurrence

Let $a_1, a_2\dots$ be a sequence of reals such that $a_1 = a_2 = 1$, and $$a_{n + 2} = \frac{a_{n + 1}^3 + 1}{a_n}$$ for $n \ge 1$. It appears to be the case that all of these values are integers. How would one show that this is the case?

For $k=2,3,4,5,6,7,8,9,10,11,12$ even $a_5$ is no integer anymore. Reconsider your conjecture.
The $a_n$ increase very fast. Up to $n=10$ they are all integers. What is interesting about them is that they are all products of very few large primes. See here: