The formula for the Bockstein $\beta:H_n(X;\mathbb{Z}/p\mathbb{Z})\to H_{n-1}(X;\mathbb{Z}/p\mathbb{Z})$ is $$\beta[c\otimes 1]=[\frac{1}{p}\partial c\otimes 1]$$ (McCleary page 456)

How about for the higher Bockstein $$\beta_r: H_n(X;\mathbb{Z}/p^r\mathbb{Z})\to H_{n-1}(X;\mathbb{Z}/p^r\mathbb{Z})$$?

I read that it is related to the connecting homomorphism, but am quite confused about the formula. Is it (just a guess based on my limited and possibly wrong understanding):

$$\beta_r[c\otimes 1]=[\frac{1}{p^r}\partial c\otimes 1]$$

I read some texts but strangely none of them seem to write the explicit formula. Thanks for any help.