Yes. Let $ k_1, ... k_{1997}$ be distinct primes. We claim that for each $ i, 1 \le i \le 1997$ we can have $ ia = m_i^{k_i}$ for some $ m_i$. Writing $ a = p_1^{e_1} p_2^{e_2} ...$ we see that the set of conditions $ ia = m_i^{k_i}$ gives, for each $ j$, a system of congruences for $ e_j$. For example, for $ i = 1$ and $ p_i = 2$ the system is $ e_1 \equiv 0 \bmod k_1$ $ e_1 \equiv - 1 \bmod k_2$ $ e_1 \equiv 0 \bmod k_3$ $ e_1 \equiv - 2 \bmod k_4$ and so forth. More generally we have $ e_j \equiv - v_{p_j}(i) \bmod k_i$ where $ v_p(i)$ is the greatest power of $ p$ dividing $ i$. By CRT each of these systems has a positive integer solution.