XXII OBM

Let $\sigma (n)$ be the sum of all positive divisors of $n$, where $n$ is a positive integer (for instance, $\sigma (6)=12$ and $\sigma (11)=12$). We say that $n$ is almost perfect if $\sigma (n)=2n-1$ (for instance, 4 is almost perfect since $\sigma (4)=7$). Let $n\mathrm{mod}k$ be the remainder of the division of $n$ by $k$ and $s(n)={\sum }_{1\le k\le n}n\mathrm{mod}k$ (for instance, $s(6)=0+0+0+2+1+0=3$ and $s(11)=0+1+2+3+1+5+4+3+2+1+0=22$). Prove that $n$ is almost perfect if and only if $s(n)=s(n-1)$.