Cube Hunting 2

Find the set of integers nn such that n6+24n3+192 n^6 + 24n^3 + 192 is a perfect cube. If there are NN distinct solutions n1,n2,,nNn_1,n_2,\ldots ,n_N, submit N+j=1Nnj\displaystyle N + \sum_{j=1}^N n_j.

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