Prime-cubes can be good too!

Call a positive integer \(N\) \(\textbf{good}\), if it is prime and the cube of the number in decimal representation can be split into 3 parts such that their sum adds up to the original number.

Let \(N_1, N_2, N_3, ..., N_n\) be all of the distinct \(\textbf{good}\) numbers on the interval \([1, 10^9]\). Determine the value of \(R\) if:

\[\sum^{n}_{i=1}N_{i}^{N_{i}}\equiv R \mod10^{10}\]

Details and Test cases:

  • \(0\) is not a positive integer.
  • Consider \(297^3= 26198073\). So, \(26+198+73 = 297\). But it is not a good number since \(297\) is not prime.

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