What is the class of real sequences satisfying these conditions?

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

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

2. 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: $u_{n+1}=u_{n}+2$.

3. 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.

• (a) What do you mean by "the class of the real sequence"? The class of all sequences with this property? (b) Any 1-element set $\omega$ will satisfy property 2. (c) "equidistribution of integer sequences mod 1" does not make sense. – Goldstern Dec 22 '16 at 22:35
• I meant what is the class of all real sequence u_k which satisfies the below conditions or property as you said " the existence of this sequence if it is possible " and in the part of the motivation of this question i meant equidistribution of real sequence not integer mayeb i have a wrong typo – zeraoulia rafik Dec 22 '16 at 22:40
• What do you mean by the final clause in condition 2? Isn't the sum taken over all $n\in\omega$? Or are you asserting that $\omega=\{n\in \mathbb{N}^*\mid u_{n+1}=u_n+2\}$? – Arturo Magidin Dec 22 '16 at 23:14
• Here the sum is taken over all n\n* with u(n+1)=u(n)+2 , as you defined it in the second formula – zeraoulia rafik Dec 22 '16 at 23:23
• So: the sum of the reciprocals of the primes diverges; the sum of the reciprocals of the twin primes converges; we don't know whether or not there are infinitely many twin primes; you want to characterize real sequences that have this in common with the primes. I hope you get lots of good answers (but if you're hoping to settle the twin prime problem this way, I'm not optimistic). – Gerry Myerson Dec 22 '16 at 23:59