Returning to the stopping time of the process (first part here):
Here is the plot of the stopping time embellished with a few logarithmic functions.
This structure is explained by looking at the number of operations that a number experiences before reaching .
No operations. Such number are obviously of the form , with stopping time . These creates the points which lie on the curve .
One operation. To find such numbers, follow their orbit backward: a series of multiplication by , then operation, then more multiplications by . This leads to numbers of the form where must be even in order for to be divisible by . The stopping time is . Since , the corresponding points lie close to the curve . Also notice that unlike the preceding case, clusters appear: there may be multiple pairs with even and the same . The larger the sum is, the more such pairs occur.
Two operations. Tracing the orbit backwards again, we find that these are numbers of the form
It is straightforward to work out the conditions on which allow both divisions by to proceed. They are: either is odd and , or is even and . In any event, the stopping time is and the number itself is approximately . On the above chart, these points lie near the curve . Clustering will be more prominent than in the previous case, because we now deal with triples that will be nearby each other if is the same.
k) operations of kind. These yield the points near the curve , or, to put it another way, . The plot above shows such curves for .