Given a set $A \subseteq \mathbb{N}$ of positive integers, put $$ S_A: \mathbb{N} \rightarrow \mathbb{R}, \ \ \ \ N \mapsto \sum_{n \in A \cap \{1, \dots, N\}} \frac{1}{n}. $$ There are obvious quite "natural" sets $A$ such that $S_A(N) = O(\log(N))$ (e.g. $A = \mathbb{N}$) and such that $S_A(N) = O(\log(\log(N)))$ (e.g. $A$ is the set of prime numbers).

Question: Which "natural" sets $A$ are there such that $S_A(N) = O(\log(\log(\log(N))))$?

By "natural" I mean: "has a short description which is not merely a trivial rewrite of the condition in the question".

Given a set $A \subseteq \mathbb{N}$ of positive integers, put $$ S_A: \mathbb{N} \rightarrow \mathbb{R}, \ \ \ \ N \mapsto \sum_{n \in A \cap \{1, \dots, N\}} \frac{1}{n}. $$ There are obvious quite "natural" sets $A$ such that $S_A(N) = O(\log(N))$ (e.g. $A = \mathbb{N}$) and such that $S_A(N) = O(\log(\log(N)))$ (e.g. $A$ is the set of prime numbers).

Question: Which "natural" sets $A$ are there such that $S_A(N) = O(\log(\log(\log(N))))$ (and not $S_A(N) = o(\log(\log(\log(N))))$, to avoid trivial answers)?

By "natural" I mean: "has a short description which is not merely a trivial rewrite of the condition in the question".