The least common multiple of $2n$ and $n^2$ will be $n^2$ if $n$ is even and $2n^2$ when $n$ is odd. First, solving the equation $n^2 = 14n-24$ yields $n = 2$ and $n = 12$. Both of which are even therefore are valid. Next, solving $2n^2 = 14n-24$ yields $n = 3$ and $n = 4$. Only $3$ is odd. Therefore, the answer is $2 + 3 + 12 = \boxed{17}$.