[seqfan] How big is "literal reading of prime factorization of n"?

[Neil Sloane]

If n = 8 = 2^3 = 2*2*2
then reading this literally we could get 23 (A080670),
or 222 (A037276) or 101010 etc.

Here is a list of different versions of the resulting sequence:
A037276, A133500, A080670, A230625, A067599, A068633, ...

We know a "random number" n has about log log n distinct prime factors,
because of the work of Hardy and Ramanujan, Erdos, Kac, etc. (the entries
A001221 = omega(n), A001222 = Omega(n) should mention these results, by the
way).

My question is, for a large "random" n, what is the average size of any of
A037276, A133500, A080670, A230625, A067599, A068633 ?

This is a notoriously tricky subject. I'm looking for theorems, not guesses!

Here's an old reference:

Hardy, G. H., and S. Ramanujan. "The normal number of prime factors of a
number n [Quart. J. Math. 48 (1917), 76–92]." *Collected papers of
Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI* (2000): 262-275.