Pratt [Pr24] has proved this is irrational, conditional on a uniform version of the prime $k$-tuples conjecture.
Tao has observed that this is a special case of [257], since \[\sum_{n\geq 2}\frac{\omega(n)}{2^n}=\sum_p \frac{1}{2^p-1}.\]
This was essentially solved by Hančl [Ha91], who proved that such a sequence needs to satisfy \[\limsup_{n\to \infty} \frac{\log_2\log_2 a_n}{n} \geq 1.\] More generally, if $a_n\ll 2^{2^{n-F(n)}}$ with $F(n)<n$ and $\sum 2^{-F(n)}<\infty$ then $a_n$ cannot be an irrationality sequence.
Kovač and Tao [KoTa24] have proved that any strictly increasing sequence such that $\sum \frac{1}{a_n}$ converges and $\lim a_{n+1}/a_n^2=0$ is not such an irrationality sequence. On the other hand, if \[\liminf \frac{a_{n+1}}{a_n^{2+\epsilon}}>0\] for some $\epsilon>0$ then the above folklore result implies that $a_n$ is such an irrationality sequence.
Kovač and Tao [KoTa24c] have proved that $2^n$ is not such an irrationality sequence. More generally, they prove that any strictly increasing sequence of positive integers such that $\sum\frac{1}{a_n}$ converges and \[\liminf \left(a_n^2\sum_{k>n}\frac{1}{a_k^2}\right) >0 \] is not such an irrationality sequence. In particular, any strictly increasing sequence with $\limsup a_{n+1}/a_n <\infty$ is not such an irrationality sequence.
On the other hand, Kovač and Tao do prove that for any function $F$ with $\lim F(n+1)/F(n)=\infty$ there exists such an irrationality sequence with $a_n\sim F(n)$.
Erdős believed that $a_n^{1/n}\to \infty$ is possible, but $a_n^{1/2^n}\to 1$ is necessary.
This has been almost completely solved by Kovač and Tao [KoTa24], who prove that such a sequence can grow doubly exponentially. More precisely, there exists such a sequence such that $a_n^{1/\beta^n}\to \infty$ for some $\beta >1$.
It remains open whether one can achieve \[\limsup a_n^{1/2^n}>1.\] A folklore result states that $\sum \frac{1}{a_n}$ is irrational whenever $\lim a_n^{1/2^n}=\infty$, and hence such a sequence cannot grow faster than doubly exponentially - the remaining question is the precise exponent possible.
A negative answer was proved by Kovač and Tao [KoTa24], who proved even more: there exists a strictly increasing sequence of positive integers $a_n$ such that \[\sum \frac{1}{a_n+t}\] converges to a rational number for every $t\in \mathbb{Q}$ (with $t\neq -a_n$ for all $n$).
Crmarić and Kovač [CrKo25] have shown that the answer to this question is no in a strong sense: for any $\alpha \in (0,\infty)$ there exists a function $f:\mathbb{N}\to\mathbb{N}$ such that $f(n)\to \infty$ as $n\to\infty$ and \[\sum_{n\geq 1} \frac{1}{(n+1)\cdots (n+f(n))}=\alpha.\] It is still possible that this sum is always irrational if $f$ is assumed to be non-decreasing; Crmarić and Kovač show that the set of the possible values of such a sum has Lebesgue measure zero.