> I can't see an infinite set coming from any reasonable heuristic. You're summing 1/n^2.

--well, I basically agree with you. I can think of a heuristic that says infinite, but I don't like that heuristic :) Meanwhile I happened to notice these summaries of immense computations:

http://oeis.org/A014224
http://oeis.org/A028491
http://oeis.org/A051783
http://oeis.org/A171381 (this last one surprised me!)

which show that Brennan & my examples are all there are,

up to 3^195430 at least (wow!) PROVIDED we only allow

ONE prime-power, the other two need to be genuine primes.

It would not be hard to use these to genuinely deal with prime powers too,

but I haven't.

In the proofs of things like the Catalan conjecture, usually they prove it for very large

numbers, then deal with the small numbers by computer.

The fact that these computations have been so immense may make it

feasible to prove there are no more examples with TWO nonprime prime-powers.