TSTST 2025 Problem 4

Let $n \geq 2$ be a positive integer. Let $a_1, a_2, \ldots, a_n$ be a sequence of positive integers such that

$$\gcd(a_1, a_2),\ \gcd(a_2, a_3),\ \ldots,\ \gcd(a_{n-1}, a_n)$$

is a strictly increasing sequence. Find, in terms of $n$, the maximum possible value of

$$\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}$$

over all such sequences.