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How fast can $a_n\to \infty$ grow if \[\sum\frac{1}{a_n}\quad\textrm{and}\quad\sum\frac{1}{a_n+1}\] are both rational?
Cantor observed that $a_n=\binom{n}{2}$ is such a sequence. If we replace $+1$ by a larger constant then higher degree polynomials can be used - for example if we consider $\sum\frac{1}{a_n}$ and $\sum\frac{1}{a_n+8}$ then $a_n=n^3+6n^2+5n$ is an example.