Use your inequalities!

Algebra Level 3

Find the least positive integer value of $n$ such that for any $n$ positive reals $a_{1},a_{2},\ldots,a_{n}$ , we have

$\dfrac{a_{1}^2+1}{a_{1}}+\dfrac{a_{2}^2+1}{a_{2}}+\cdots +\dfrac{a_{n}^2+1}{a_{n}} \geq 2016.$