Parabola

Let $\mathrm{S}$ be the set of all $\mathrm{a} \in \mathrm{N}$ such that the area of the triangle formed by the tangent at the point $\mathrm{P}(\mathrm{b}$, c), b, c $\in \mathbb{N}$, on the parabola $y^{2}=2 \mathrm{a} x$ and the lines $x=\mathrm{b}, y=0$ is $16 $ unit², then $\sum\limits_{\mathrm{a} \in \mathrm{S}} \mathrm{a}$ is equal to :