It is well-known that

A: The series of the reciprocals of the primes diverges

My question is whether property A is in some sense a truth strongly tied to the nature of the prime numbers.

For instance, can you give an example of an infinite subset $A \subseteq \mathbb{N}$ such that $\sum_{a \in A} \frac{1}{a}$ diverges, $A$ doesn't contain any prime numbers and the $k$-th member of $A$ is greater than the $k$-th prime for infinitely many $k$?

Property A tells us that the primes are a rather fat subset of $\mathbb{N}$. Is there a way to define a topology on $\mathbb{N}$ such that every dense subset of $\mathbb{N}$ (under this topology) corresponds to a fat subset of the natural numbers?

What do you think about this?