Question 168
Let $n \ge 2$ be a positive integer. Let $a_1, a_2, \dots, a_n$ be a sequence of positive integers such that
$$
\gcd(a_1, a_2), \gcd(a_2, a_3), \dots, \gcd(a_{n-1}, a_n)
$$
is a strictly increasing sequence. Find, in terms of $n$, the maximum possible value of
$$
\frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}.
$$
over all such sequences.
$$
\gcd(a_1, a_2), \gcd(a_2, a_3), \dots, \gcd(a_{n-1}, a_n)
$$
is a strictly increasing sequence. Find, in terms of $n$, the maximum possible value of
$$
\frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}.
$$
over all such sequences.