Comments by "EebstertheGreat" (@EebstertheGreat) on "Untouchable Numbers - Numberphile" video.

5 is the only odd untouchable number if a slight strengthening of Goldbach's conjecture holds. Goldbach's conjecture states that ever even number greater than 2 is the sum of two prime numbers. A stronger statement that also seems true is that every even number greater than 6 is the sum of two distinct prime numbers. If this is true, then given any odd number n > 7, we can write n= p + q + 1 with p and q distinct primes. But the only proper factors of pq are 1, p, and q, so its aliquot sum is s(pq) = 1 + p + q = n. That leaves the special cases of 1, 3, 5, and 7. For any prime p, s(p) = 1, s(4) = 3, and s(8) = 7. So only 5 is untouchable.
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There are no other ways to collapse. To hit 0, it first has to hit 1 (unless it started at 0), and to hit 1, it has to first hit a prime (unless it started at 1). So every sequence that terminates at 0 goes a -> b -> ... -> p -> 1 -> 0 with p prime. More specifically, p will always be an odd prime other than 5 (since 2 and 5 are untouchable), unless you start at p. And to answer your question, the asymptotic density of abundant numbers is known to be between 0.2474 and 0.2480. That means that as n increases without bound, the proportion of numbers less than n that are abundant approaches some limit that is about 24.77%.
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@Phlosioneer That's true, but no counterexamples are known (greater than 6). It's just a stronger version of Goldbach's strong conjecture.
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