Problem

Source: 2024 Turkey TST P8

Tags: number theory



For an integer $n$, $\sigma(n)$ denotes the sum of postitive divisors of $n$. A sequence of positive integers $(a_i)_{i=0}^{\infty}$ with $a_0 =1$ is defined as follows: For each $n>1$, $a_n$ is the smallest integer greater than $1$ that satisfies $$\sigma{(a_0a_1\dots a_{n-1})} \vert \sigma{(a_0a_1\dots a_{n})}.$$Determine the number of divisors of $2024^{2024}$ amongst the sequence.