RIOPLATENSE 1993 Problem 5

Prove that for every integer $k \ge 2$ there are $k$ different natural numbers $n_1$ , $n_2$ , $...$ , $n_k$ such that: $$\frac{1}{n_1}+\frac{1}{n_2}+...+\frac{1}{n_k}=\frac{3}{17}$$