MEMO 2011 Problem 8

We call a positive integer $n$ amazing if there exist positive integers $a, b, c$ such that the equality $$n = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab)$$ holds. Prove that there exist $2011$ consecutive positive integers which are amazing.Note. By $(m, n)$ we denote the greatest common divisor of positive integers $m$ and $n$ .