VN-TST 1997 Problem 3

Find the greatest real number $\alpha$ for which there exists a sequence of infinitive integers $(a_n)$ , ( $n = 1, 2, 3, \ldots$ ) satisfying the following conditions:

1. $a_n > 1997n$ for every $n \in\mathbb{N}^{*}$ ;
2. For every $n\ge 2$ , $U_n\ge a^{\alpha}_n$ , where $U_n = \gcd\{a_i + a_k | i + k = n\}$ .