The time complexity for the merge sort algorithm is $T(n) = 2T(n/2)+\Theta(n)$, and the solution to this recurrence is $\Theta(n\lg n)$.
However; assume you are not dividing the array in half for sorting, but instead in, say, $a$ pieces. Then the time would be something like $T(n) = aT(n/a) + \Theta(n \log_2 a)$. (The last term being the time to perform divide-and-merge for $a$ pieces with a total size of $n$. For positive, integer $a$, what is the solution to this recurrence? (assuming $T(n) = 1$ for $n = 1$ and that $n=a^x$ where x is a positive integer as well).
I have a hunch it would probably be $\Theta(n \log_2 n)$ for any $a$, but I'm not really sure how to prove it, with the logarithm being in a different base than $2$ .