RU-SPB 2011 Problem 102

$n$ - some natural. We write on the board all such numbers $d$ , that $d\leq 1000$ and $d|n+k$ for some $1\leq k \leq 1000$ . Let $S(n)$ -sum of all written numbers. Prove , that $S(n)<10^6$ and $S(n)>10^6$ has infinitely many solutions.