smc

Let the roots be $r,s,r+s$, and let $t=rs$. Then $(x-r)(x-s)(x-(r+s))$ $=x^3-(r+s+r+s)x^2+(rs+r(r+s)+s(r+s))x-rs(r+s)=0$ and by matching coefficients, $2(r+s)=2004⟹r+s=1002$. Then our polynomial looks like $$x^3-2004x^2+(t+1002^2)x-1002t=0$$ and we need the number of possible products $t=rs=r(1002-r)$. Because $m=t+1002^2$ is an integer, we also note that $t$ must be an integer. Since $r>0$ and $t>0$, it follows that $0<t=r(1002-r)<501^2=251001$, with the endpoints not achievable because the roots must be distinct and positive. Because neither $r$ nor $1002-r$ can be an integer, there are $251,000-500=\boxed{250,500}$ possible values of $n=-1002t$.