Let $A$ consist of the integers of type $a_n=[n \log n \log \log n]$, (say for $n \geq 20$). Then $\sum_{a_n \leq x} \frac{1}{a_n} \sim \log \log \log x$. This follows by partial summation or by considering the integral $\int 1/(t \log t \log \log t)\, dt$.
As this sequence might be considered to be a trivial reformulation, here is an example related to primes:
Let $A_b$ denote the sequence of primes, whose sum of digits is also prime, in a fixed base $b$.
The sum of digits has typically size about $\frac{b-1}{2}\log n$, and the probability that this number is prime
is about $\frac{c_b}{\log \log n}$. Hence the density of this sequence is about $\frac{1}{\log n \log \log n}$, and $\sum_{a_n\leq x} \frac{1}{a_n}\sim C_b \log \log \log x$. The details can be found in a paper by Glyn Harman (Theorem 2).
Counting Primes whose Sum of Digits is Prime,
https://cs.uwaterloo.ca/journals/JIS/VOL15/Harman/harman2.html