I am working on the following problem.
Consider the function $B:\mathbb{N}\to\mathbb{R}$ defined by: $$B(n) = \begin{cases} 1 & \text{if $n\leq 2$,}\\ 3\cdot B(\lceil n/\log_2 n\rceil) + n & \text{if $n\geq 3.$} \end{cases} $$
Find an explicit function $g(n)$ such that $B(n) \in \Theta(g(n)).$
I can see that $n\in O(B(n))$, and intuitively it seems as though $B(n)\in O(n)$, too.
However, I am not sure how to rigorously show this. Furthermore, the Master Theorem cannot be directly applied in this case.
I would appreciate suggestions as to how to solve this problem.