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I am now confident the answer to the question is no; the basic reason is that the function $r(m)=\sigma(m)/m$ "can't grow fast enough" to support the existence of such a number. I will attempt an argument. Gerhard "Sleeping On It Is Good" Paseman, 2015.07.15
There are also multiperfect numbers. I suspect there is a proof that metaperfect numbers (infinitely iterated form of multiperfects) do not exist. Gerhard "Likes Using The Term 'Metaperfect'" Paseman, 2015.07.14
I found the Bennett and Potts reference online, and they have effectively answered 3) and 4), except for the enumeration bit. I still think this approach deserves further reflection, but I am not first on the scene. There are connections to Costas arrays and other combinatorics to. Also, I haven't verified my "payoff" yet, so I may have to retract. Gerhard "Will Make The Computer Count" Paseman, 2015.07.14
You may be able to use something like (z+c)^n for a well chosen complex c and integer n for the map, and show that (z+c)^n - z "has large roots". Gerhard "Not Ready For Such Complexity" Paseman, 2015.07.13
Although I do not have a proof, there should be something (Stern-Brocot tree?) that says using the mediant iteratively to approximate a real by a rational gives at each stage the fraction with smallest denominator closest to the real: you don't miss any fractions with smaller denominator in doing so. I feel pretty confident about my prediction as a result. Once you have a/b comfortably in the interval with b not much smaller than $1/\epsilon$, tweaking top and bottom to get nearest primes should work. Gerhard "Look! I'm About To Fly!" Paseman, 2015.07.13
Indeed, Golomb and Taylor in dealing with a more specific construct (hexagonal arrays) have considered this bijection and related problems, and refer to an older paper. Although I would appreciate other references answering 3) or 4), I still hope for more attempts at enumeration. Gerhard "Just Asking For The World" Paseman, 2015.07.13
It seems that one reference is arxiv.org/abs/0911.2384 on Honeycomb arrays. I would appreciate learning more, as I think question 4) still has not been answered regarding enumeration. Gerhard "Still Learning This Internet Thing" Paseman, 2015.07.12
This is essentially a prime gap conjecture: For what primes q can we guarantee a prime p with xq < p < (x+e)q? Granted we don't know that there is a prime greater than xq and within log(xq)loglog(xq), but I would say it is likely once xq gets above 100. So pick q large enough so that (log(xq)^2)/q smaller than epsilon. Gerhard "I'd Say You're Probabilistically Safe" Paseman, 2015.07.11
@Roland, I probably did at one point. I thank you for reminding me anyway. I encourage you to remind me of other things, as I am unsure what will prove significant. Gerhard "Seeks Many Forms Of Closure" Paseman, 2015.07.11
I see a potential danger in formalism: loss of creativity. Mathematicians aren't always aware of the strength of the proof system needed to make a formal argument; they often are aware of the words needed to attempt communicating ideas and something resembling a convincing argument. If people spend a lot of time trying to formalize a proof so that someone else can explain it to a machine, that will be time taken away from creating new ideas and new realms for mathematicians to explore. Gerhard "Stays Off The Formalistic Bandwagon" Paseman, 2015.07.06
