Euler Phi Function of Prime Power/Corollary

Corollary to Euler Phi Function of Prime Power
Let $\phi: \Z_{>0} \to \Z_{>0}$ be the Euler $\phi$ function.

Then:
 * $\map \phi {2^k} = 2^{k - 1}$

Proof
We have that:
 * $1 - \dfrac 1 2 = \dfrac {2 - 1} 2 = \dfrac 1 2$

It follows from Euler Phi Function of Prime Power:
 * $\map \phi {2^k} = \paren {\dfrac 1 2} 2^k = 2^{k - 1}$