Inspired by misreading MEMO 2014 T7.
We call a sequence of $n$ distinct non-negative integers $a_1, a_2, \dots a_n$ meanly if it satisfies the following condition:
For every $k\leq n$, we have: $$\frac{1}{k}\sum_{j=1}^k{a_j}\in\mathbb{N}$$ Find the minimum element sum of a meanly sequence of integers of length $n$.
Comments by @Griboullis suggest that the optimal sequence is connected to OEIS A293688 for the sum and A002251 for the $a_j$, which appear linked to upper and lower Wythoff sequences.