Melvyn Knight once asked  for which integers is there an integer triple so that .
For example, 11 has two such representations: and .
Solutions for this problem can be found using elliptic curves . This Demonstration shows a sample solution for all solvable values from to . When one solution exists, there are an infinite number of solutions.
 H. Pfoertner. "Integers n representable as the product of the sum of three nonzero integers with the sum of their reciprocals." The On-Line Encyclopedia of Integer Sequences. (Jan 24, 2023) oeis.org/A085514.