Symplifying a sum with multiple indices IM trying to understand the following simplification 
$$ \sum_{k,n,m} [k^3 \leq n < (k+1)^3 ][n=km][1 \leq n \leq 1000] = 1 + \sum_{k,m} [k^3 \leq km < (k+1)^3][1 \leq k <10] $$
where [P(x)] = 1 if $P(x)$ is true statement a $0$ otherwise. Why is the above true? Im having hard time trying to compute few terms of the sum. For example, if $n=1$, then km=1
$$ \sum_{k,n,m} [k^3 \leq n < (k+1)^3 ][1=km][1 \leq n \leq 1000]  $$
but, do we simplify this? Im very confused. Any help would be greatly appreciated.

The necessary and sufficient condition for $n$ to exist is 
$$(k+1)^3\gt 1\quad\text{and}\quad 1000\ge k^3\iff 1\le k\le 10$$
So, we have
$$\sum_{k,n,m}[k^3\le n\lt (k+1)^3][n=km][1\le n\le 1000]$$
$$=\sum_{k,n,m}[k^3\le n\lt (k+1)^3][n=km][1\le n\le 1000][1\le k\le 10]\tag1$$
Separating this sum into two cases, the case where $k=10$ and the case where $1\le k\lt 10$, we have
$$\begin{align}&(1)=\sum_{k,n,m}[k^3\le n\lt (k+1)^3][n=km][1\le n\le 1000][\color{red}{k=10}]\\\\&\qquad +\sum_{k,n,m}[k^3\le n\lt (k+1)^3][n=km][1\le n\le 1000][1\le k\color{red}{\lt} 10]\\\\&=\sum_{n,m}[10^3\le n\lt 11^3][n=10m][1\le n\le 1000]\\\\&\qquad +\sum_{k,n,m}[k^3\le n\lt (k+1)^3][n=km][1\le n\le 1000][1\le k\lt 10]\\\\&=\sum_{n,m}[n=1000][m=100]+\sum_{k,m}[k^3\le km\lt (k+1)^3][1\le km\le 1000][1\le k\lt 10]\\\\&=1+\sum_{k,m}[k^3\le km\lt (k+1)^3][1\le k\lt 10]\end{align}$$
