Wanna Cube Up?

How many 18-digit numbers are there such that the sum of the cubes of their digits is a perfect cube?

Details and Assumptions

  • As an explicit example, 6151 is a 4-digit number such that the sum of the cubes of its digits is a perfect cube: \(6^3+1^3+5^3+1^3=343=7^3\).
  • There are a total of 57 4-digit numbers that satisfy this property.

This is a harder version of the problem Wanna square up?!.

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