Re: n, 2n-1, 2n+1 all prime or prime-power (maybe n-2 also)
> 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/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.