On simultaneous rationality of two Ahmes series

Paul Erdős asked how rapidly a sequence of positive integers $(n_k)$ can grow if both series $\sum_k 1/n_k$ and $\sum_k 1/(n_k-1)$ have rational sums. In this note we show that there exists an exponentially growing sequence $(n_k)$ with this property. Previous records had polynomial growth, even for easier variants of the problem, regarding the series $\sum_k 1/n_k$ and $\sum_k 1/(n_k-d)$ for any concrete nonzero integer $d$. Moreover, using the same ideas we negatively answer another irrationality question, posed by Paul Erdős and Ronald Graham. Namely, there exists a bounded sequence of positive integers $(b_k)$ such that $\sum_k 1/(2^k+b_k)$ is a rational number.