Divisor game

Alice and Bob invented a two-player game. At first, Alice chooses a positive integer $k$ in the range from 1 to some fixed integer $n$. Then Bob asks questions of the form ‘Is $k$ divisible by $m$?’, where $m$ is a positive integer. Alice answers each such question with ‘yes’ or ‘no’. Bob wants to know what number Alice has in mind by asking as few questions as possible. Your task is to write a program that plays the game as Bob.

Let $d(n)$ denote the minimal number of questions that have to be asked to find $k$, regardless of what $k$ Alice chooses (for given $n$). Your program’s answer for a test case will be considered correct if $k$ is correctly determined using no more than $d(n)$ questions.