Total Number of Bits Needed to Represent a List of N elements
In the RAM model of computation, machine words are $\Theta(\log n)$ bits long (where $n$ is the size of the input in bits), and operations on machine words can be done in $O(1)$. Counting sort has running time $O(N + k)$ in the RAM model assuming that each value is $\Theta(\log n)$ bits long.

Let us now consider an array consisting of $N$ values, the maximal value of which is $k$. We would need $\log k$ bits to store each value. Assuming that we use $\Theta(\log k)$ bits per value, the input length is $n = \Theta(N\log k)$ bits, and so a machine word is $\log N + \log\log k + O(1)$ bits long. Let us denote by $w$ the number of machine words needed to store a single entry, which we can calculate by
$$ w = \left\lceil \frac{\log k}{\log N + \log\log k} \right\rceil = \Theta\left(\max\left(1,\frac{\log k}{\log N + \log\log k}\right)\right). $$
The running time of counting sort is $O(Nw + k)$.