Find three integers in Harmonic Progression

It is difficult to think of many triplets which are in HP also having the condition that all of them are integers. Is there a way of systematically defining them all or a procedure to find more than a few examples? Some examples are 3,4,6 and 4,6,12. Are there even other triplets possible ?

I also found out if a, b, c are in GP then a+b,2b,b+c are in HP. This generates more of them but I would like to have a more generalized idea.

Solution 1:

Take any triple in arithmetic progression: for example, $8,9,10$.

Take their reciprocals; in reverse order, if you want the result to be increasing. In the example: $\frac1{10}, \frac19, \frac1{8}$.

Multiply through by the least common denominator: $\frac{360}{10}, \frac{360}{9}, \frac{360}{8}$ or $36, 40, 45$.

You can scale them up by any other factor you like: $360, 400, 450$ are also in harmonic progression.

This gets you all the possible harmonic triplets, because you can do all these steps again and get the starting arithmetic triplet back. So every harmonic triplet comes from an arithmetic one.