Erdős Problem #266 — Shifted Reciprocal Series

Is the following universal assertion false? For every sequence $a:\mathbb{N}\to\mathbb{N}$ such that $a(n)\geq1$ for every $n$ and

$$\sum_{n\in\mathbb{N}}\frac1{a(n)}$$

converges, there exists a natural number $t\geq1$ such that

$$\sum_{n\in\mathbb{N}}\frac1{a(n)+t}$$

is irrational. Equivalently, does there exist a sequence $a:\mathbb{N}\to\mathbb{N}$ with $a(n)\geq1$ for every $n$ and convergent $\sum_{n\in\mathbb{N}}1/a(n)$ such that, for every natural number $t\geq1$, the series $\sum_{n\in\mathbb{N}}1/(a(n)+t)$ is rational?