Solution:

(ans. 49
$(2 a_{1}+1)(2 a_{2}+1) \cdots(2 a_{k}+1)$ where $1000=p_{1}^{a_{1}} p_{2}^{a_{2}} \cdots p_{k}^{a^{k}}=2^{3} 5^{3} ;$ so 49. Let $\frac{x y}{x+y}=n \Rightarrow x y-n x-n y=0 \Rightarrow(x-n)(y-n)=n^{2} \Rightarrow x>n, y>n$. In the factorization $n^{2}=p_{1}^{2 a_{1}} p_{2}^{2 a_{2}} \cdots p_{k}^{2 a^{k}}$ each divisor of $n^{2}$ determines a solution, hence the answer.)