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

[Dean Hickerson]

I've extended the list of solutions of  (d^3+e^3+f^3)/(d e f) = n  up to
max(d,e,f) = 28393.  It's still at:

     http://www.math.ucdavis.edu/~dean/defn.txt

The comment for sequence A072716 says:

> It is known that any of the numbers below does not appear in the
> sequence: (a) multiples of 4, (b) numbers congruent to 7 mod 8,
> (c) numbers of the form 2^m*k + 3 with odd numbers m >= 3 and k >= 1.

Does anyone here know how to prove that?

What can be said about the non-integer values of  (d^3+e^3+f^3)/(d e f)?

Also, based on my calculations it appears that the values of d, e, and f
can't be divisible by 4, but I don't see how to prove that either.  Can
all non-multiples of 4 occur as d, e, or f?  (29, 30, and 51 are the first
ones that I haven't seen yet.)

Dean Hickerson
dean at math.ucdavis.edu