Rigour Requested

[davieddy]

Probability of a factor between x and x+dx is dx/(xlnx)
Let P(x) be the probability of no factor <x.
Then P(x+dx) = P(x)(1-dx/(xlnx))
dP/P = -dx/(xlnx)
Integrating from x=a^X to x=a^Y gives the probability of no factors in this range = X/Y, from where we get the familiar probability of one or more factors between 2^X-1 and 2^X = 1/X.

I am grateful to Oliver for the following sanity check:
The product of all prime factors of N is N.
The sum of their logs is logN.
Expected sum of their logs is the integral of logx(1/xlnx)dx from 1 to N = logN.

******
All such simple stuff that there must be some truth in it.
However there are two large elephants in the room named "independence" and "2kp+1".
Can someone get rid of them please?

David