Question 44
If $k$ is an integer, let $\mathrm{c}(k)$ denote the largest cube that is less than or equal to $k$. Find all positive integers $p$ for which the following sequence is bounded:
$$
a_{0}=p \quad \text{ and } \quad a_{n+1}=3 a_{n}-2 c\left(a_{n}\right) \quad \text{ for } n \geqslant 0
$$
$$
a_{0}=p \quad \text{ and } \quad a_{n+1}=3 a_{n}-2 c\left(a_{n}\right) \quad \text{ for } n \geqslant 0
$$