Question 109
For an integer $n$, let $\sigma(n)$ be the sum of all positive divisors of $n$. A sequence of positive integers $(a_i)_{i=0}^{\infty}$ is defined as follows: $a_0 = 1$ and for each $n \ge 1$, $a_n$ is the smallest integer greater than 1 such that
$$
\sigma(a_0 a_1 \cdots a_{n-1}) \mid \sigma(a_0 a_1 \cdots a_n).
$$
Determine the number of terms of this sequence dividing the number $2024^{2024}$.
$$
\sigma(a_0 a_1 \cdots a_{n-1}) \mid \sigma(a_0 a_1 \cdots a_n).
$$
Determine the number of terms of this sequence dividing the number $2024^{2024}$.