A conjecture about number theory and tetration

[tommy1729]

A conjecture about number theory and tetration.

A natural number n may be factored as n = p1^a1 p2^a2 p3^a3 ... where the p_m are distinct prime numbers and a_m are natural numbers. Since the a_m 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.
This problem is about the height of n which we denote as h(n).
We Define:

D_n = lim N -> +oo N^-1 * |{k =< N : h(k) => n}|

D_n is sort of the density of numbers with height at least n. It is obvious that D_1 = 1 since all numbers have height at least 1.

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 afaik [Andrew Snowden has found them] are :

1.42333 < a < 1.4618

ANDREW SNOWDEN CONJECTURE :

for all n :
1/2 < D_n 2^^n =< 3

I felt the need to show the conjecture to give it more attention as it deserves.

Also some recent members might not have seen it yet and that would be a pity if they would never see it.

I also would like to see more work concerning this , for instance how Andrew found 1.42333.

If I am very optimistic it might lead to a new method for tetration.

I said very optimistic ...

regards

tommy1729