This post won't make much sense unless my previous posts on this topic are read:
- Odds and Evens on June 17th 2021
- Attractors, Vortices and Captives on June 18th 2021
- Binary Odds and Evens on June 19th 2021
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Figure 1 |
Figure 1 shows the sums of odd and even digits in the number systems from 10 down to 2. It also shows the ratio between the two sums. With even bases (10, 8, 6, 4 and 2), it can be seen that the sum of odd digits are larger than the sum of even digits. With odd bases (9, 7, 5 and 3), the situation is reversed.
In my previous post on Binary Odds and Evens, it was apparent that no vortices were possible and thus there were only captives and attractors. In base 10, captives could be captured by attractors and vortices. This same situation should prevail in the bases from 9 down to 3. However, the main focus of this post is to look at the first 100,000 integers and enumerate them according to the nomenclature that I have developed. What I discovered is that there are:
- 3725 attractors
- 58977 captives of these attractors
- 914 vortices with a total of 3975 vorticals
- 34223 captives of these vortices
Figure 1 shows a graphical representation of this data:
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| Figure 1: link |
The average number of vorticals in a vortex is almost exactly four. The maximum size of a vortex in the range chosen is 11 and there are two of these:
- 81191, 81193, 81195, 81197, 81199, 81201, 81203, 81204, 81205, 81207, 81211
- 18211, 18191, 18193, 18195, 18197, 18199, 18201, 18203, 18204, 18205, 18207
The minimum size is of course is two and there are many of these e.g. 198 --> 200 --> 198.
Here is a permalink to the SageMathCell program that calculated this information. I did need to do a fair bit of tinkering to get it all to work but I'm very pleased with the final result.
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