None of the Harmonic numbers $H_n = \sum_{k=1}^n 1/k$ are integers for $n>1$
(e.g., [this MSE question and answer](https://math.stackexchange.com/q/2746/237)).

>**Q**. Define the $r$-th root Harmonic number $H_n^{1/r} = \sum_{k=1}^n 1/{k^{1/r}}$.
E.g., for $r=2$, $H_n^{1/2} = \sum_{k=1}^n 1/\sqrt{k}$.
Is $H_n^{1/r}$ ever an integer for $n>1$ and $r \ge 2$ an integer?

Of course there are many close calls, e.g.,
$H_{202}^{1/2} \approx 27.0002$,
$H_{1132}^{1/3} \approx 162.00004$,
$H_{222}^{1/4} \approx 76.000003$.
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**Answered** in comments by Vesselin Dimitrov: ***No***.
In fact all these sums are irrational.