Question 149
Let $n \gt 2$ be a positive integer. Consider all numbers $S$ of the form
$$
S = a_1 a_2 + a_2 a_3 + \dots + a_{k-1} a_k,
$$
with $k \gt 1$, and $a_i$ being positive integers such that $a_1 + a_2 + \dots + a_k = n$. Determine all numbers that can be represented in the given form.
$$
S = a_1 a_2 + a_2 a_3 + \dots + a_{k-1} a_k,
$$
with $k \gt 1$, and $a_i$ being positive integers such that $a_1 + a_2 + \dots + a_k = n$. Determine all numbers that can be represented in the given form.