In the arrangement shown spring, pulleys, strings are ideal. The natural frequency of the system for vertical oscillations is

From the figure it is clear that, if the displacement of block is $x_0$, extension in spring will be $4x_0$

Force on the block will be $\left(8k{x}_0\right)2=16k{x}_0$

$$\begin{gathered} \Rightarrow 16k{x}_0=k_{eq}{x}_0, \Rightarrow 16k{x}_0=k_{eq}{x}_0 \\ \Rightarrow \omega =\sqrt{\frac{k_{eq}}{m}}=\sqrt{\frac{16k}{m}}=4\sqrt{\frac{k}{m}} \\ t=\frac{\omega }{2\pi }=\frac{2}{\pi }\sqrt{\frac{k}{m}} \end{gathered}$$