A perfect cube is \(n = k^3\) for some integer \(k\). Sequence: 0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.
Sums of cubes are notoriously hard. Fermat's Last Theorem (proven by Wiles in 1995) tells us no positive cubes sum to a cube. The taxicab number 1729 is the smallest expressible as the sum of two positive cubes in two different ways. As recently as 2019, mathematicians finally found a way to write 42 as a sum of three (signed) cubes — it had been the last number under 100 for which no such representation was known.