We call a set of positive integer good, if the greatest common divisor of all of the elements in this set is $1$. $a_n$ is the number of good subsets of $\{1,2,...,n\}$. Find all integer $n \ge 2019$, such that, $\forall \> 0<k<n, k \in Z$, $\frac{a_{n+k}}{a_n} < \frac{a_n}{a_{n-k}}$
Is there a quick solution to this problem?
What I thought: $2^n-1= \sum_{d=1}^{n}{a_{[\frac{n}{d}]}}$a
2019 NGC Girls MO