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It feels odd the sum of the first example is 26. (All the number nerds, please forgive me). It's such a usual number.

From my read, it probably flows from 26 = 27 - 1 = 3^3 - 1, given the equation is looking at cubes.

It's not. (Law of Small Numbers)

As the exponent k in the denominator grows, the sum is approximately k! (factorial)