Is it an integer?

Number theory
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Tolaso J Kos
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Is it an integer?

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Post by Tolaso J Kos »

Investigating something on the serieses with Harmonic numbers the following occured. I do not have a solution , yet I would like to see one.

Let $n \geq 1$ and denote as $\mathcal{H}_n$ the $n$ - th harmonic number. Let us also denote as ${\rm lcm} ( \cdot , \cdot )$ the least common multiple. Is it true that the sequence defined as

$$a_n=n^{\mathcal{H}_n {\rm lcm}(1,2,\dots,n)}$$

is an integer, that is contains also integer numbers?
Imagination is much more important than knowledge.
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