Numeric Residue Set Values for the Descent from 445 to 27

These values are taken from an rsetprog run, and are presented here in predecessor order, which is the reverse to their appeance in the program's output.

First, the routine rsetidea recursively develops the l.d.a.  from e to ebsssbssbsbbssssbs,  storing the values of its progress, here called the outer path, in mylist as shown in columns 1-3.  These values are simply retained until needed for output by densout, which prints them completely unchanged. 

The subroutine densout is invoked when the leaf node is encounted.  It scans mylist,  from leaf node to header node,  calculating the c[d] values encountered in the Collatz trajectory,  here called the inner path,  for output as shown in columns 4 and 5 . Thus two versions of the residue sets are produced for each node in the l.d.a.

     steps taken for      outer path      inner path           instantiation
     descent to 27       {c[d]}values     {c[d] values        d(5)/d(3)  as 3^i
                   e      5          8    445   1836660096    229582512   --
                  eb     17         32    593   1377495072     43046721   16
                 ebs     11         64    395    918330048     14348907   15
                ebss      7        128    263    612220032      4782969   14
               ebsss    175        256    175    408146688      1594323   13
              ebsssb    233       1024    233    544195584       531441   12
             ebsssbs    155       2048    155    362797056       177147   11
            ebsssbss    103       4096    103    241864704        59049   10
           ebsssbssb    137      16384    137    322486272        19683    9
          ebsssbssbs     91      32768     91    214990848         6561    8
         ebsssbssbsb    121     131072    121    286654464         2187    7
        ebsssbssbsbb    161     524288    161    382205952          729    6
       ebsssbssbsbbs    107    1048576    107    254803968          243    5
      ebsssbssbsbbss     71    2097152     71    169869312           81    4
     ebsssbssbsbbsss     47    4194304     47    113246208           27    3
    ebsssbssbsbbssss     31    8388608     31     75497472            9    2
   ebsssbssbsbbssssb     41   33554432     41    100663296            3    1
  ebsssbssbsbbssssbs     27   67108864     27     67108864            1    0

The c[d] values in the outer path are all canonical. Only the last one in the inner path is canonical. The residue sets in the outer path contain the elements in the residue sets appearing in the inner path.  Specifically: 263 is the second instantiation of 7[128] (zero origin indexing is in use!). 395 is the sixth instantiation of 11[64], 593 is the eighteeth instantiation of 17[32] and 445 is the fifty-fifth instantiation of 5[8]. This divergence of the outer path c-values from those of the inner path is quite typical among lengthy l.d.a.s.

The d-values of the inner path are successive powers of three, owing to the multiplication by 3 during all Collatz trajectories. The sixth column reflects the progressive diminution of the density contributions of the successive elements of the l.d.a. in the inner as compared to the outer paths as the d-values ratios indicate.  The seventh column shows those ratios as powers of 3. The minute contribution which the nodes of the descent to 27 make to the total density of the integers in this (and all lengthy l.d.a.s) is evident from the large d values (in column 5) associated with every node traversed in the Collatz trajectory in such deeply developed predecessor trees.

The appearance of various levels of higher instantiations of residue sets in the elaboration of the prdecessor tree even for the first instantiation of an l.d.a. is well illustrated in this example. But higher instantiaions of any l.d.a. designate the occurrences of elements of the infinite set of residue set members. An entire l.d.a. can be extended to high instances by applying the transofrmation to each of its elements. Thus, the first (right after the zeroth) instantiation of the descent to 27 can be calculated by adding the values in column 5 to those of column 4,  and the second instantiation by adding twice the value in column 5 to that in column 4.  Clearly successive instantiations will be far more widely scattered among the large integers than the initial instantiation is among small integers. This effect contributes to the difficulty of recognizing the predecessor tree's structure by examining any form of integer-based predecessor trees

The first four steps in this are l.d.a. headers themselves, as may be seen on another page which describes the rsetidea program. The following table verifies that the deeper residue sets are members of the four cited subtrees of the predecessor tree.

e         5*         8     5[8]
eb       17*        32     1[8]
ebss     11*        64     3[8]
ebsss     7*       128     7[8]
        175        256     7[8]
        233       1024     1[8]
        155       2048     3[8]
        103       4096     7[8]
        137      16384     1[8]
         91      32768     3[8]
        121     131072     1[8]
        161     524288     1[8]
        107    1048576     3[8]
         71    2097152     7[8]
         47    4194304     7[8]
         31    8388608     7[8]
         41   33554432     1[8]


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