Example of an element not lying in the $A_p$ class but in a slightly bigger class.

Question

Let $\Omega=\mathbb{R}^N$ and $p>1$.

Define the class $B_p=\{w(x): w\in L^1_{loc}(\Omega); w^{-\frac{1}{p-1}}\in L^1_{loc}(\Omega)\}$ and consider the standard Muckenhoupt class of weights $A_p$.

Intuitively, from the definition I think the set $B_p\setminus A_p$ is non-empty. If so, Can You give and example of an element in $B_p\setminus A_p$?