On a previous page we noticed that except for 27, log (Max(N))/log(N) doesn't appear to exceed 2. Though I can't prove this is true I can assume it and use it as the basis of a backwards searching algorithm for numbers that converge on a particular root. For instance if I want to find all the numbers than converge on 85 that are less than 1024 I can reject sequences that pass through numbers greater than 1024 squared (1048576).
Applying this algorithm gives us the following results;
Start (1 value):
[85]
First iteration (7 values):
[113, 453, 1813, 7253, 29013, 116053, 464213]
2nd (19 values):
[75, 301, 1205, 4821, 19285, 77141, 308565, 2417, 9669, 38677, 154709, 618837, 4835, 19341, 77365, 309461, 154737, 618949, 309475]
3rd (39 values):
[401, 1605, 6421, 25685, 102741, 410965, 803, 3213, 12853, 51413, 205653, 822613, 25713, 102853, 411413, 51427, 205709, 822837, 1611, 6445, 25781, 103125, 412501, 51569, 206277, 825109, 103139, 412557, 3223, 12893, 51573, 206293, 825173, 103153, 412613, 206307, 825229, 825265, 412633]
The results are summarised in the table below.
This first chart shows the number of entries less than 1024 as a function of tree depth (green cells in the table). The second chart shows the width of the search tree, for odd numbers up to 1048576.
The bad news is that although the furthest value found was only 8 iterations out the search itself didn't stop until after 61.
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The 85 Tree: See a an illustration of the base of the 85 tree. |
In the table below the green cells show a search out to 1024. Unshaded values, up to 1048576, show the range of numbers we had to search. Adding up the tree width column will show the number of numbers we tested - 7582. Testing the 511 values less than 1024 would have been quicker.
To get numbers out to 4096 this algorithm requires 118564 possibilities be explored, for 8192 this grows to 468857, just to find the first 99 values that converge on 85. This chart shows the distribution of those 99 values in terms of the number of odd-hops from the root.
As we double the depth to search we roughly quadruple the set of values to search.
So it's not looking that good an algorithm.