The following heuristic probabilistic argument supports the
Conjecture (see ).
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
Furthermore it suggests that the total stopping time
is (in some average sense) a constant multiple of .
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