Cantor observed that $a_n=\binom{n}{2}$ is such a sequence. If we replace $-1$ by a different constant then higher degree polynomials can be used - for example if we consider $\sum_{n\geq 2}\frac{1}{a_n}$ and $\sum_{n\geq 2}\frac{1}{a_n-12}$ then $a_n=n^3+6n^2+5n$ is an example of both series being rational.

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.