A finite set of positive integers
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
is called
meanly if for each of its nonempty subsets the arithmetic mean of its elements is also a positive integer. In other words, A is meanly if
![\frac{1}{k}(a_1 + \ldots + a_k)](/media/m/0/9/d/09d915c9a017f0e013d62a4d8d852ed8.png)
is an integer whenever
![k \geq 1](/media/m/4/7/4/474e08320ac4dfd51b6214797b6d06be.png)
and
![a_1, \ldots, a_k \in A](/media/m/d/8/1/d8103ae83ea86e171a783e5945bf7d7f.png)
are distinct.
Given a positive integer
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
, determine the least possible sum of the elements of a meanly
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
-element set.
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A finite set of positive integers $A$ is called [i]meanly[/i] if for each of its nonempty subsets the arithmetic mean of its elements is also a positive integer. In other words, A is meanly if $\frac{1}{k}(a_1 + \ldots + a_k)$ is an integer whenever $k \geq 1$ and $a_1, \ldots, a_k \in A$ are distinct.
Given a positive integer $n$, determine the least possible sum of the elements of a meanly $n$-element set.