Super Attractors

In my own private terminology, I deem a number an odd-even attractor if its sums of odd digits and even digits are the same. I use the term attractor because numbers that are not attractors are "attracted" to such numbers. For example, let's take the case of 134. Here the sum of the odd numbers is 1 + 3 = 4 and the sum of the even numbers is 4. Thus it is an odd-even attractor. 

Let's take a number like 122 that is not an odd-even attractor. The sum of the even digits (4) exceeds the sum of the odd digits (1).  The difference between odd and even digits is 1 - 4 = -3 and this will be added to the original number to get 119. Now the sum of the odd numbers (11) exceeds that of the non-existent even numbers (0) and this is added to 119 to get 130. Repeating the process we get 134 which is an attractor.

In this system, I've chosen to subtract the sum of the even digits from the sum of the odd digits. This is quite arbitrary and I could have chosen to subtract the sum of the odd digits from the even digits but for odd-even or even-odd attractors this doesn't matter. A similar system can be adopted for prime and non-prime digits. The prime digits are 2, 3, 5 and 7 whereas the non-prime digits are 0, 1, 4, 6, 8 and 9. A number wherein the sum of the prime digits equals that of the non-prime digits is called, in my nomenclature, a prime-non-prime attractor. An example would be 358 where 3 + 5 = 8.

A number that is not a prime-non-prime attractor is 356. Here the sum of prime digits is 8 and the sum of the non-prime digits is 6. We chose to subtract the sum of non-prime digits from the sum of the prime digits to get 2 which we add to 356 to get 358 which is a prime-non-prime attractor.

The question that I was interested in is how many numbers are both odd-even attractors and prime-non-prime attractors? We might term these super attractors. In the range up to 40000, there are 222 such numbers (permalink) and they are:

112, 121, 211, 336, 358, 363, 385, 538, 583, 633, 835, 853, 1012, 1021, 1102, 1120, 1201, 1210, 2011, 2101, 2110, 3036, 3058, 3063, 3085, 3306, 3360, 3445, 3454, 3467, 3476, 3508, 3544, 3580, 3603, 3630, 3647, 3674, 3746, 3764, 3805, 3850, 4345, 4354, 4367, 4376, 4435, 4453, 4534, 4543, 4556, 4565, 4578, 4587, 4637, 4655, 4673, 4736, 4758, 4763, 4785, 4857, 4875, 5038, 5083, 5308, 5344, 5380, 5434, 5443, 5456, 5465, 5478, 5487, 5546, 5564, 5645, 5654, 5667, 5676, 5748, 5766, 5784, 5803, 5830, 5847, 5874, 6033, 6303, 6330, 6347, 6374, 6437, 6455, 6473, 6545, 6554, 6567, 6576, 6657, 6675, 6734, 6743, 6756, 6765, 6778, 6787, 6877, 7346, 7364, 7436, 7458, 7463, 7485, 7548, 7566, 7584, 7634, 7643, 7656, 7665, 7678, 7687, 7768, 7786, 7845, 7854, 7867, 7876, 8035, 8053, 8305, 8350, 8457, 8475, 8503, 8530, 8547, 8574, 8677, 8745, 8754, 8767, 8776, 10012, 10021, 10102, 10120, 10201, 10210, 11002, 11020, 11200, 12001, 12010, 12100, 20011, 20101, 20110, 21001, 21010, 21100, 30036, 30058, 30063, 30085, 30306, 30360, 30445, 30454, 30467, 30476, 30508, 30544, 30580, 30603, 30630, 30647, 30674, 30746, 30764, 30805, 30850, 33006, 33060, 33600, 34045, 34054, 34067, 34076, 34405, 34450, 34504, 34540, 34607, 34670, 34706, 34760, 35008, 35044, 35080, 35404, 35440, 35800, 36003, 36030, 36047, 36074, 36300, 36407, 36470, 36704, 36740, 37046, 37064, 37406, 37460, 37604, 37640, 38005, 38050, 38500

Figure 1 shows the rather uneven distribution of such numbers in the range up to 40000:

Figure 1

All of the numbers greater than 10000 contain the digit 0. Attractors are very much base-specific and thus fall into the realm of recreational mathematics. The big gaps occur between 12100 and 20011 and 21100 and 30036. Numbers that are not attractors of either sort but are close to super attractors do not necessarily end up attracted to the nearest attractor. 

Take 30035 that is next to the super attractor 30036. Here is its prime-non-prime trajectory:

\(30035 \rightarrow 30046 \rightarrow 30039 \rightarrow 30036 \rightarrow 30036\)

While it ends up at the nearby super attractor, the same is not true when subjected to the odd-even trajectory:

\(30035 \rightarrow 30046 \rightarrow 30039 \rightarrow 30054 \rightarrow 30058 \rightarrow 30058\)

It ends up at the more distant super attractor 30058.

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.