Find a sequence of positive integers $f(n)$ ($n \in \mathbb{N}$) such that: (i) $f(n) \leq n^8$ for any $n \geq 2$; (ii) for any distinct $a_1, \cdots, a_k, n$, $f(n) \neq f(a_1) + \cdots+ f(a_k)$.
Source: Romania TST 1995 Test 3 P4
Tags: number theory
Find a sequence of positive integers $f(n)$ ($n \in \mathbb{N}$) such that: (i) $f(n) \leq n^8$ for any $n \geq 2$; (ii) for any distinct $a_1, \cdots, a_k, n$, $f(n) \neq f(a_1) + \cdots+ f(a_k)$.