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".