RO-TST 2015

Given an integer $N \geq 4$ , determine the largest value the sum $$\sum_{i=1}^{\left \lfloor{\frac{k}{2}}\right \rfloor+1}\left( \left \lfloor{\frac{n_i}{2}}\right \rfloor+1\right)$$ may achieve, where $k, n_1, \ldots, n_k$ run through the integers subject to $k \geq 3$ , $n_1 \geq \ldots\geq n_k\geq 1$ and $n_1 + \ldots + n_k = N$ .