Prove there exists $2011$ consecutive amazing integers Recently, I have found this problem: 

We call a positive integer $n$ amazing if there exists 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 exists $2011$ consecutive positive integers which are amazing.

Here some amazing numbers:

In the picture from left to right you find the numbers: $n$, $a$, $b$ and $c$.
I have tried to solve this problem in a lot of different ways, for example using the definition of $GCD$, or divisibility but I can't go on. Any idea?
Note:by $(m, n)$ we denote the greatest common divisor of positive integers $m$
and $n$.

Claim; For any $j \in \mathbb{N}$ multiples of $j^3+2j^2$ are amazing.
Proof; Set $a=j, b=jk, c =j^2 k$;
One has that $(b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab) = k(j^3+2j^2)$.
Now construct a sequence $c_d, 1 \leq d \leq 2011, c_d \in \mathbb{N}$ with each $c_d$ of the form $j^3+2j^2$ also with the condition that $gcd(c_u,c_v) = 1$ for $u \neq v$. 
All that is remaining to do now is to solve the following congruence;
$m \equiv 0$ $ mod $ $c_1$
$m +1 \equiv 0$ $mod$ $c_2$
.
.
.
$m +2010 \equiv 0$ $mod$ $c_{2011}$
