Is it an integer?
Posted: Wed Aug 30, 2017 6:50 pm
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?
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?