A number theory problem by Akshat Sharda

Number Theory Level 5

Let $a=\dfrac{(n+3)^3}{n-1}$ where $a$ and $n$ are integers.

Let the integral values of $n$ satisfying the above equation be $n_{1},n_{2},n_{3}, \ldots, n_{m}$ .

Find $\left(\displaystyle \sum^{m}_{p=1}n_{p}\right)+m$ .