RO-TST 1996 Problem 8

Let $p_1,p_2,\ldots,p_k$ be the distinct prime divisors of $n$ and let $a_n=\frac {1}{p_1}+\frac {1}{p_2}+\cdots+\frac {1}{p_k}$ for $n\geq 2$ . Show that for every positive integer $N\geq 2$ the following inequality holds: $\sum_{k=2}^{N} a_2a_3 \cdots a_k <1$

Laurentiu Panaitopol