Question 80
We consider $A$, the set of all positive integers with exactly $2019$ positive integer divisors, and, for each $n \in A$, we denote
$$
S_n = \frac{1}{d_1 + \sqrt{n}} + \frac{1}{d_2 + \sqrt{n}} + \dots + \frac{1}{d_{2019} + \sqrt{n}},
$$
where $d_1, d_2, \dots, d_{2019}$ are the positive divisors of $n$.
Determine the maximum value of $S_n$ when $n$ runs through the set $A$.
$$
S_n = \frac{1}{d_1 + \sqrt{n}} + \frac{1}{d_2 + \sqrt{n}} + \dots + \frac{1}{d_{2019} + \sqrt{n}},
$$
where $d_1, d_2, \dots, d_{2019}$ are the positive divisors of $n$.
Determine the maximum value of $S_n$ when $n$ runs through the set $A$.