Question 65
Let $n$ be a positive integer. Find the number of all finite strictly increasing sequences $a_{0}=1, a_{1}, \ldots, a_{k}=2.3^{n}$ of positive integers with the following property: $\prod_{i=1}^{k}\left[\frac{a_{i}+a_{i-1}-1}{a_{i-1}}\right]=2.3^{n}$, where $[x]$ is the integral part of $x$.