Let $A \subset \mathbb{N}^{\ast}$. Exhibit $(u_n)_{n \geq 1}$ such as $\forall q \in \mathbb{N}^{\ast}$, $\sum_{n\geq 1} u_n^q$ diverges iff $q \in A$

Question

I am having difficulties with the following exercise on numerical series :

Let $A \subset \mathbb{N}^{\ast}$. Exhibit a sequence $(u_n)_{n \geq 1} \in \mathbb{C}^{\mathbb{N}^{\ast}}$ such as $\forall q \in \mathbb{N}^{\ast}$, $\sum_{n\geq 1} u_n^q$ diverges iff $q \in A$.

I tried first to study the case of $A$ being a singleton, but I do not have any relevant result yet.

I have no idea how to go about it ...

How can I proceed?

Thank you very much for your help