Define harmonic numbers for a complex argument $z$ as $H_z=\frac{\Gamma'(z+1)}{\Gamma(z+1)}-\Gamma'(1)$.

For $n\in\mathbb{N}$, $H_n$ are usual harmonic numbers $\sum^n_{k=1} k^{-1}$ . They are obviously rational and are known (Taeisinger 1915) to be non-integers for $n>1$.

*Question:* Is there a non-integer rational $q$ such that $H_q\in\mathbb{Q}$?