Question 66
Find all natural numbers $n$ for which there exist $a_1, a_2, \dots, a_n \in \mathbb{N}$ such that
$$
a_1 + 2a_2 + \dots + n a_n = 6n
$$
and
$$
\frac{1}{a_1} + \frac{2}{a_2} + \dots + \frac{n}{a_n} = 2 + \frac{1}{n}.
$$
$$
a_1 + 2a_2 + \dots + n a_n = 6n
$$
and
$$
\frac{1}{a_1} + \frac{2}{a_2} + \dots + \frac{n}{a_n} = 2 + \frac{1}{n}.
$$