The Modest Magnetism of 26596

Today's diurnal number, 26596, has the interesting property that the sums of its odd and even digits are equal. Thus we see that 5 + 9 = 14 and 2 + 6 + 6 = 14. I've written extensively about the implications of this odd-even algorithm in the following posts:

• Odds and Evens on June 17th 2021
• Attractors, Vortices and Captives on June 18th 2021
• Binary Odds and Evens on June 19th 2021
• Odds and Evens: Statistics on June 21st 2021

Figure 1: ring magnet

I use the term attractor to describe numbers like 26596 that are unchanged by the odd-even algorithm. They act like magnets, attracting other numbers to them as these numbers are subjected to repeated applications of the algorithm. I've applied the term "modest magnetism" to 26596 because it attracts only 10 numbers while 26569, with exactly the same digits, manages to attract 92 numbers. These captured numbers I've termed, appropriately, captives.

The captives of 26596 and their trajectories are as follows:

• 26579 --> [26579, 26592, 26596]
• 26585 --> [26585, 26579, 26592, 26596]
• 26590 --> [26590, 26596]
• 26591 --> [26591, 26598, 26596]
• 26592 --> [26592, 26596]
• 26594 --> [26594, 26596]
• 26598 --> [26598, 26596]
• 26603 --> [26603, 26592, 26596]
• 26605 --> [26605, 26596]
• 26613 --> [26613, 26603, 26592, 26596]

Figure 2 shows the same information in a more pictorial way. It can be seen that some numbers are only one step removed (26590, 26592, 26594, 25598, 26605), others are two steps removed (26591, 26579, 26603) and others are three steps removed (26585, 26613):

Figure 2

There are 11 attractors in the range between 26500 and 26700, representing 5.5% of the total numbers in the range. These attractors are:

• 26503 with no captives
• 26525 with no captives
• 26530 with 19 captives
• 26547 with no captives
• 26552 with 18 captives
• 26569 with 92 captives
• 26574 with no captives
• 26596 with 10 captives
• 26659 with no captives
• 26677 with no captives
• 26695 with no captives

It can be seen that seven of the attractors have no captives. They might be termed inert attractors. Clearly some centuries have more attractors than others. Between 26500 and 26600, there are eight attractors but between 26600 and 26700 there are only three (and all of them inert). Anyway, the above list provides a basis for comparison between 26596 and other attractors in a range of roughly 100 on either side of it.

For some time now, I've been keeping a daily check on what numbers are attractors and what numbers are captives of attractors. Figure 3 shows the trajectory for 26595, my diurnal age yesterday:

Figure 3

Interestingly tomorrow's number, 26597, forms part of a vortex together with 26610. This vortex manages to capture 13 numbers: 26599, 26611, 26612, 26614, 26616, 26617, 26618, 26623, 26625, 26627, 26633, 26637, 26639. As can be seen, some captives are not captured by attractors but by vortices like {26597, 26610}. This is all explained in my paper that I uploaded to Academia: link.