It is "common knowledge" that 13 + 23 + ... + n3 = (1 + 2 + ... + n)2.
Hence the set of numbers {1,2,...,n} has the property that the sum of its cubes is the square of its sum. Are there any other collections of numbers with this property? Yes, and the following method is guaranteed to generate such a set.
Pick a number, any number. Did I hear you say 63? Fine.
List the divisors of 63, and for each divisor of 63, count the number of
divisors it has:
63 has 6 divisors (63, 21, 9, 7, 3, 1)
21 has 4 divisors (21, 7, 3, 1)
9 has 3 divisors (9, 3, 1)
7 has 2 divisors (7, 1)
3 has 2 divisors (3, 1)
1 has 1 divisor (1).
The resulting collection of numbers has the same cool property. Namely
63 + 43 + 33 + 23 + 23 + 13 = 324 = (6+4+3+2+2+1)2.