I am interested in having an upper bound for the cardinality of 


$\#\left\{n\leq x\,:\quad\omega(n)=k, \omega(n+2)=\ell\right\}$
for $k,\ell\geq 1$, 


where $\omega(n)=\sum\_{p\vert n}1$ counts the number of (distinct) prime factors of the integer $n$.  

I would actually like a sharp upper bound (up to a constant) for $k=\ell=\log\log x$. 
I thought this was known, but I can't find anything in the literature. For $k\ll\sqrt{\log\log x}$ and $\ell\ll\log\log x$ I can have a good upper bound, but I can't reach $k=\ell=\log\log x$.


Does anyone has a reference or an idea please ? 


Thank you very much !



Independence between the number of prime factors of $n$ and $n+2$
The problem as been solved and [generalized](https://arxiv.org/pdf/1710.04877.pdf). Thread can be closed.


