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]