The Erdos-Woods Problem

Setup

Fundamental Theorem of Arithmetic: Every integer $n >1$ can be expressed uniquely as a product $$n = p_1^{e_1} \cdots p_k^{e_k}$$ where the $p_i$ are prime number satisfying $p_i < p_j$ if $i < j$ and the $e_i$ are positive integers.

Defintion (Radical): The radical of an integer $n$, denoted $\text{rad}(n)$, is defined simply as the product of all prime divisors of $n$. That is, if $n = p_1^{e_1} \cdots p_k^{e_k}$ then $\text{rad}(n) = p_1 \cdots p_k.$

Remark: The only reasonable thing to do is to define $\text{rad}(1) = 1.$

The Evil Wizard

Suppose you are confronted by an evil wizard who presents you with the following challenge:

"I'm going to pick a positive integer $n > 1$ and then I shall tell you the radical of each integer from $n$ to $n+k$. Then you must tell me what $n$ is."

For which $k$ should you accept this challenge? Discuss...