Conjecture (see [28]).
Pick an odd integer 
at random and iterate the function T
until another odd integer 
occurs.
Then 
of the time 
of the time 
of the time 
, and so on.
If one supposes that the function T is sufficiently ``mixing''
that successive odd integers in the trajectory of n
behave as though they were drawn at random
from the set of odd integers
for all k, then the
expected growth in size between two consecutive odd integers in such a
trajectory is the multiplicative factor
Consequently this heuristic argument suggests that on average the iterates in
a trajectory tend to shrink in size, so that divergent trajectories should not
exist.
Furthermore it suggests that the total stopping time
is (in some average sense) a constant multiple of 
.
(Click here for more.)
From the viewpoint of this heuristic argument, the central difficulty of the
problem lies in understanding in detail the ``mixing'' properties of
iterates of the function 
for all powers of 2.
The function 
does indeed have some ``mixing'' properties given by
Theorems B and K below;
these are much weaker than what one needs to settle the 
Conjecture.
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