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Much less is known about the total stopping time function than about the
stopping time function.
One phenomenon immediately observable from a table of the total
stopping times of small integers is the occurrence of many pairs and triples of
integers having the same finite total stopping time.
From Figure 1 we see that
,
,
,
,
and
.
Indeed for larger values of **n**, multiple consecutive values occur with the
same total stopping time.
For example there are 17 consecutive values of **n** with
for .
A related phenomenon is that over short ranges of **n** the function
tends to assume only a few values
(C. W. Dodge [70]).
As an example the values of for
are given in Table 3.
Only 19 values for are observed, for which a
frequency count is given in Table 4.
Both of these phenomena have a simple explanation;
they are caused by coalescence of trajectories of different **n**'s after a
few steps.
For example the trajectories of **8k + 4** and **8k + 5** coalesce after 3 steps,
for all .
More generally, the large number of coalescences of numbers and
close together in size can be traced to the trivial
cycle (1,2),
as follows.
Suppose and have ,
and let .
Then the trajectories of and coalesce after at most
iterations, since
, since the trajectory of
continues to cycle around the trivial cycle.
If in addition , which nearly always
happens if and are about
the same size,
then the trajectories of ,
and coalesce after at most
iterations, for .
In particular, then holds for .
In this case the original coalescence of and has produced an
infinite arithmetic progression of coalescences.
The gradual accumulation of all these arithmetic progressions of
coalescences of numbers close together in size leads to the phenomena observed
in Tables 3 and 4.
Although the Conjecture asserts that all integers **n** have a finite
total stopping time, the strongest result proved so far concerning the
density of the set of integers with a finite total stopping time is
much weaker.

TABLE 3. Values of the total stopping time for .

TABLE 4. Values of and their frequencies for .

### Theorem G

(Crandall). * Let*
* Then there is a positive constant such that
for all sufficiently large ***x**. (Click
here for the current best
result.)

Assuming that the Conjecture is true, one can consider the problem
of determining the * expected size*
of the total stopping time function
.
Crandall [28] and Shanks [63] were guided by
probabilistic heuristic arguments (like the one described earlier)
to conjecture that the average order of
should be a constant times ; more precisely, that

( Click here and
here for further information on
.)
A modest amount of empirical evidence supports these conjectures, see [28].

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