Harmonic Triple

A triple  of positive integers satisfying  is said to be harmonic if

In particular, such a triple is harmonic if the reciprocals of its terms form an arithmetic sequence with common difference  where

One can show that there exists a one-to-one correspondence between the set of equivalence classes of harmonic triples and the set of equivalence classes of geometric triples where here, two triples  and  are said to be equivalent if , i.e., if there exists some positive real number  such that .

REFERENCES:

VanderBurgh, I. (Ed.). "Mathematical Mayhem: Mayhem Solutions." Crux Math. 36, 141-143, 2010.