OBM 2000 Problem 2

Let $s(n)$ be the sum of all positive divisors of $n$ , so $s(6) = 12$ . We say $n$ is almost perfect if $s(n) = 2n - 1$ . Let $\mod(n, k)$ denote the residue of $n$ modulo $k$ (in other words, the remainder of dividing $n$ by $k$ ). Put $t(n) = \mod(n, 1) + \mod(n, 2) + \cdots + \mod(n, n)$ .

Show that $n$ is almost perfect if and only if $t(n) = t(n-1)$ .