CN-LEAGUE 1999 Problem 3

$n$ is a given positive integer, such that it's possible to weigh out the mass of any product weighing $1,2,3,\cdots ,ng$ with a counter balance without sliding poise and $k$ counterweights, which weigh $x_ig(i=1,2,\cdots ,k),$ respectively, where $x_i\in \mathbb{N}^*$ for any $i \in \{ 1,2,\cdots ,k\}$ and $x_1\leq x_2\leq\cdots \leq x_k.$ $(1)$ Let $f(n)$ be the least possible number of $k$ . Find $f(n)$ in terms of $n.$ $(2)$ Find all possible number of $n,$ such that sequence $x_1,x_2,\cdots ,x_{f(n)}$ is uniquely determined.