Timeline for Do $r$-th root Harmonic numbers ever sum to integers?

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Apr 13, 2017 at 12:19	history	edited	CommunityBot		replaced http://math.stackexchange.com/ with https://math.stackexchange.com/
Aug 9, 2014 at 10:41	history	edited	Joseph O'Rourke	CC BY-SA 3.0	added 104 characters in body
Aug 9, 2014 at 2:56	comment	added	Vesselin Dimitrov		We may also consider real irrational exponents $\lambda = 1/r \in \mathbb{R}$. For a given $\lambda \in \mathbb{R} \setminus \mathbb{Q}$, are there only finitely many $n$ for which $H_n^{\lambda}$ is an integer? (a rational?) Is there a uniform bound on the number of exceptions? Surely the answer is "yes," the point is whether it is possible to prove this or not. It was a Putnam contest problem that unless $\lambda$ is a (non-positive) integer, not all $H_n^{\lambda}$ can be integers.
Aug 9, 2014 at 1:56	comment	added	Vesselin Dimitrov		They are always irrational. Quite generally, you may show that for any positive rationals $a_i \in \mathbb{Q}^{> 0}$ and any positive integers $k_i \in \mathbb{N}$, a finite sum $\sum_1^n \sqrt[k_i]{a_i}$ is always irrational unless it is rational term-by-term.
Aug 9, 2014 at 1:34	history	asked	Joseph O'Rourke	CC BY-SA 3.0