I'm interested in finding the class of the real sequences $u_{k}$, $k\in \mathbb{N^*}$ which satify the following conditions:

$\displaystyle \sum_{k=1}^{\infty}\frac{1}{u_{k}}$ diverges i.e $\displaystyle \sum_{k=1}^{\infty}\frac{1}{u_{k}}=+\infty$

There exist a non-empty set $\omega $ such that the sum $$ \sum_{n\in \omega}(\frac{1}{u_{n}}+\frac{1}{u_{n}+2})$$ has a finite positive value, i.e. there exist a positive real number $\alpha$ such that $\displaystyle \sum_{n\in \omega}(\frac{1}{u_{n}}+\frac{1}{u_{n}+2})=\alpha>0$. Here the sum is taken over all $n\in \mathbb{N^*}$ such that: $u_{n+1}=u_{n}+2$.

It is unknown whethere $\omega$ is finite or infinite .

**Note:** The motivation of the question is to study equidistribution modulo 1 of real sequences and to show if the latter has infinite elements

Thank you for any help.