a/b + b/c + c/a = n

[Hans Havermann]

> Now we can look for integer values of  a/b + b/c + c/a  more 
> efficiently:
> Look for triples (d,e,f) with  1 <= d <= e <= f,  relatively prime in 
> pairs,
> such that  d^3 + e^3 + f^3  is divisible by  d e f.  For each such 
> triple, we
> get two triples (a,b,c):  (e^2 f, f^2 d, d^2 e)  and  (e f^2, d f^2, e 
> d^2).

Is that second triple correct? Anyways, I did a quick implementation of 
this taking {d, e, f} up to 1000:

{3, 5, 6, 9, 10, 13, 14, 17, 18, 19, 21, 26, 29, 30, 38, 41, 51, 53, 
54, 57, 66, 69, 83, 86, 94, 105, 106, 126, 149, 154, 161, 166, 174, 
178, 195, 201, 209, 230, 237, 243, 250, 261, 269, 294, 323, 326, 329, 
366, 405, 446, 451, 478, 489, 534, 581, 622, 629, 630, 633, 681, 726, 
734, 789, 846, 905, 966, 978, 1011, 1097, 1410, 1491, 1658, 1713, 1718, 
1725, 1769, 1875, 1893, 2163, 2309, 2369, 2378, 2681, 3974, 6318, 7061, 
10995, 13971, 14803, 18014, 20778, 21529, 59802}

> Incidentally, the first few values of n are given by sequence A072716:
>
> %S A072716 3,5,6,9,10,13,14,17,18,19,21
> %N A072716 Integers expressible as (x^3 + y^3 + z^3)/xyz with positive
>            integers x, y, and z. (Alternatively, the integers 
> expressible
>            as x/y + y/z + z/x with positive integers x, y, and z.)
> ...
> %A A072716 Tadaaki Ohno (t-ohno(AT)hyper.ocn.ne.jp), Aug 07 2002
>
> Apparently the author doesn't know if 22 is in the sequence.

Is there a simple rule the relates the maximum {d, e, f} to how many of 
the found terms might be sequential?