Question 82
For which positive integers $n$ can $n$ positive integers $a_1, a_2, \dots, a_n$ be found such that:
$$
a_1 + a_2 + a_3 + \dots + a_n = 2n \quad \text{and}
$$
$$
a_1 + 2a_2 + 3a_3 + \dots + n a_n = n^2 ?
$$
$$
a_1 + a_2 + a_3 + \dots + a_n = 2n \quad \text{and}
$$
$$
a_1 + 2a_2 + 3a_3 + \dots + n a_n = n^2 ?
$$