第 019 题 / 100

设正整数 $n = {a}_{0} \cdot {2}^{0} + {a}_{1} \cdot 2 + \cdots + {a}_{k - 1} \cdot {2}^{k - 1} + {a}_{k} \cdot {2}^{k}$ ，其中 ${a}_{i} \in \{ 0,1\}$ ，记 $\omega \left( n\right) = {a}_{0} + {a}_{1} + \cdots + {a}_{k}$ . 则 ( )
A. $\omega \left( {2n}\right) = \omega \left( n\right)$
B. $\omega \left( {{2n} + 3}\right) = \omega \left( n\right) + 1$
C. $\omega \left( {{8n} + 5}\right) = \omega \left( {{4n} + 3}\right)$
D. $\omega \left( {{2}^{n} - 1}\right) = n$