There is a problem posted on a wall of the math club at the
University of Maryland,
by the author *Andrew Snowden*. He apparently teaches
both there and at Princeton,
and poses many interesting problems in mathematics.
The problem set was simply called *Some More Problems*, and contained
problems from algebra and number theory to real and complex analysis.
The second problem posted cought my attention, as it related
to nested exponentials, so I will reproduce it here:

A natural number `n` may be factored as
where the
are distinct prime numbers and
are natural numbers. Since the
are natural numbers, they may be factored in such a manner as well. This process may be continued, building a "factorization tree" until all the top numbers are 1. Thus any question that can be asked of trees (i.e. the height of a tree, the number of nodes in a tree, etc.) may be asked of our natural number `n`. This problem is about the height of `n` which we denote `h`(`n`). Define:

is sort of the density of numbers with height at least `n`. It is obvious that
since all numbers have height at least 1.

- Show that
- Let
`a`be the average height of a natural number (i.e. if you were to pick many numbers at random their height would average out to`a`). Using the previous part and other methods, give bounds on`a`. The best bounds [Andrew Snowden has] are1.42333 < `a`< 1.4618