Dual Attractors

I've written about attractors, vortices, vorticals and captives in many earlier posts. In my nomenclature attractors can be prime/non-prime or odd/even:

• a prime/non-prime attractor has sums of prime digits and non-prime digits that are equal
• an odd/even attractor has sums of odd and even digits that are equal

An example of a prime/non-prime attractor would be 28330 where 2 + 3 + 3 = 8 + 0. Other numbers do not have this balance and under the recursion:

number --> number + sum of prime digits - sum of non-prime digits

some will be "attracted" to 28330, meaning that repeated application of the recursion will lead to the attractor. In the case of 28330, there are 19 such numbers (termed captives):

28327, 28331, 28332, 28333, 28334, 28335, 28336, 28337, 28338, 28339, 28340, 28341, 28342, 28343, 28344, 28345, 28346, 28348, 28349

An example of an odd/even attractor would be 29612 where 9 + 1 = 2 + 6 + 2. Other numbers again do not have this balance and under the recursion:

number --> number + sum of odd digits - sum of even digits

some will be "attracted" to 29612, meaning that repeated applications of the recursion will lead to the attractor. In the case of 29612, there are 20 such numbers (termed captives):

29517, 29537, 29559, 29571, 29583, 29585, 29587, 29590, 29591, 29592, 29594, 29596, 29598, 29605, 29610, 29611, 29613, 29614, 29616, 29618

Some attractors can be both prime/non-prime and odd/even and in the range up to 40000 there are 223 of them (permalink):

0, 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

Let's take 37640 as an example:

• it is a prime attractor since prime digits 3 + 7 = 6 + 4 + 0 (non-prime digits)
  
  It has 14 captives: 37612, 37615, 37617, 37623, 37627, 37632, 37633, 37641, 37642, 37643, 37644, 37646, 37648, 37649
  
  
• it is an odd/even attractor since odd digits 3 + 7 = 6 + 4 + 0 (even digits)
  
  It has 11 captives: 37611, 37617, 37619, 37629, 37633, 37641, 37642, 37643, 37644, 37646, 37648

Of course the order of digits makes no difference and so many of these 223 numbers are just permutations of another's digits and 0's can be added anywhere because they do not affect the sum. Take 358 as an example. Permutations of its digits with or without 0 added include 385, 538, 583, 835, 853, 3058, 3085, 3508, 3580, 3805, 3850 etc.

In fact if we strip out the zeroes and put the digits in ascending order then the above list of 223 reduces to merely 112, 336, 358, 3445, 3467, 4556, 4578, 5667, 6778.

Revisiting the Odd (-) and Even (+) Algorithm

Naturally, having revisited the odd (+) and even (-) algorithm and using Gemini to generate new code, the next step was to revisit the odd (-) and even (+) algorithm. In this latter algorithm, the sum of the even digits is added to the number while the sum of the odd digits is subtracted. In both cases, the attractors remain the sum but the vortices and captives will differ. Here a permalink to the SageMathCell code that will catalog the numbers from 0 to 40000. Here is a link to the code that Gemini created. I've put the output in a Google document for later reference.

Gemini asked if I'd like to generate a comparison table for the two algorithms. Here a permalink to the algorithm that it created and a link to the code in Gemini itself. Figure 1 shows the output for the range up to 40000. I also asked about the discrepancy in the count of attractors as they should be the same. Here is a link to Gemini's explanation.

Figure 1

As can be seen, even though the attractors are the same in both cases, the number of captives that they gather can be vastly different. The attractor 8987 claims 617 captives (the record) with odd (-) and even (+) but none at all with odd (+) and even (-).

Revisiting the Odd (+) and Even (-) Algorithm

I've written extensively about this algorithm in previous posts and even uploaded a PDF to Academia (link). For some reason I decided to reread this PDF and this motivated me to get Gemini Pro 3.0 to try its hand at writing some Python code to implement this algorithm across a chosen range of numbers. Of course, I'd already done this previously using SageMath but my algorithm timed out on SageMathCell above 100,000 and I thought that any code that Gemini created would be far more efficient than any code that I could write. I tested it out on SageMathCell for a range up to one million but it still timed out. However, on a range up to 100,000, it only took a few seconds. I tried to run the code in my Jupyter notebook using the range up to one million but it spat the dummy. No problem, I'm mainly interested in the range up to 40,000 given my focus on my diurnal age. Here is a permalink to the algorithm on SageMathCell,

Here was the prompt that I gave Gemini:

Implement the following program in Python. Here are the details:

\( \textbf{Odd Even Algorithm} \)

\( \textbf{The Basic Algorithm:}\)

Let’s describe the basic algorithm first. It takes 0 or any positive integer as input, computes the sum of the number’s odd digits and the sum of the number’s even digits. The sum of the number’s odd digits is added to the number while the sum of the number’s even digits is subtracted. This process is repeated until a stable number is reached, meaning the sums of odd and even digits are equal, OR a loop is entered. 

\( \textbf{Nomenclature:}\)

I’m choosing to call stable numbers (sums of odd and even digits are equal) ATTRACTORS because under the algorithm the trajectory of many numbers will lead to such an attractor. 0 is the first such attractor and 112 is the next.

Numbers that lead back to themselves I’m calling VORTICALS. An example is 11 because its trajectory is 11, 13, 17, 25, 28, 18, 11. Similarly 13 is a vortical because its trajectory is 13, 17, 25, 28, 18, 11, 13. All these numbers lead back to themselves and collectively I call this collection of verticals a VORTEX. It can be represented as [11, 13, 17, 25, 28, 18] but any vortical in it can be placed first and only the cyclic order needs to be preserved.

Numbers whose trajectories lead to a vortex or an attractor are called CAPTIVES. 9 is a captive of a vortex because its trajectory is 9, 18, 11, 13, 17, 25, 28, 18 leads it to the vortex [18, 11, 13, 17, 25, 28]. 

\( \textbf{Applying the algorithm to a range of numbers:}\)

I’m interested in selecting a range of numbers (let’s say from 0 to 100,000) and applying the algorithm to each number in this range. What I want to keep track of are:

• Display list of attractors and their total number

• Display list of the captives of each attractor and the number of these captives for each

• Display a ranking of the attractors in order of number of captives (highest to lowest)

• Display list of vortices (plural of vortex) together with the vorticals that comprise them

• Display the captives of each each vortex and how many captives each has

• Display of ranking of vortices in order of number of captives (highest to lowest)Display overall statistics: number of attractors, number of vorticals, number of captives of attractors, number of captives of vortices.

Gemini carried these instructions out perfectly as the implemented code revealed. Here is a link to its response. I ran the program using a restricted range up to 40,000 and copied the output to a Google document (link). I can now search this document to find details concerning an attractor or vortex. For example, consider these forthcoming attractors: 28019, 28037, 28055, 28073 and 28091. Here are the results concerning their number of captives:

• 28019 has three captives (27951, 27971, 27993)

• 28037 has no captives

• 28055 has no captives

• 28073 has no captives

• 28091 has 138 captives (28100, 28102, 28104, 28106, 28108, 28110, 28111, 28112, 28113, 28114, 28115, 28116, 28117, 28118, 28121, 28123, 28127, 28129, 28139, 28140 ... and 118 more)

This of course is a very useful tool and I'll be making use of it for my diurnal number investigations. This was the summary generated for numbers in the range up to 40000:

• Total Attractors Found: 1527

• Total Vortices Found: 428

• Total Unique Vorticals: 1451

• Total Captives of Attractors: 20417

• Total Captives of Vortices: 16610

Blast From The Past

Looking back over my old tweets on Twitter (as it was once called), I noticed that my first reference to a day count occurred on August 11th 2013. See Figure 1.

Figure 1

It's a fairly unimpressive tweet that, from a mathematical perspective, simply notes that 23506 has four distinct prime factors. I wasn't aware of the Online Encyclopedia of Integer Sequences or OEIS back then, not Numbers Aplenty or most other resources that I now use. The OEIS informs us that 23506 is a member of A029793:

 A029793    Numbers \(k\) such that \(k\) and \(k^2\) have the same set of digits.

This is because \(23506^2=552532036\) and there are not many numbers in the range up to 40000 that have this property. The numbers that do are: 

0, 1, 10, 100, 1000, 4762, 4832, 10000, 10376, 10493, 11205, 12385, 12650, 14829, 22450, 23506, 24605, 26394, 34196, 36215

From Numbers Aplenty we learn that 23506 has properties that make it

• a self number because there is no number that added to its sum of digits gives 23506
  
• an untouchable number because it is not equal to the sum of proper divisors of any number

My own algorithm tells me that 23506 is:

• a xenodrome in base 9: 35217 and in base 10: 23506 because all digits are different
  
• an attractor because its sum of even and odd digits are the same (8)

These are just a few of the special properties that 23506 possesses. I've come a long way since those early days.

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.

Prime and Non-Prime Digit Sequence

I've posted extensively about the trajectories formed by repeatedly adding the sum of the odd digits of a number to the number itself while subtracting the sum of the even digits, or vice versa. My most recent post was on this topic was titled Revisiting Odds And Evens. Instead of dealing with odd and even digits, trajectories involving prime and non-prime digit sums can be considered. The prime digits are 2, 3, 5 and 7 while the non-prime digits are 0, 1, 4, 6, 7 and 9.

For example, the number associated with my diurnal age today is 27632 and it has the following trajectory of 3 steps under the prime and non-prime algorithm (permalink):

27632 --> 27640 --> 27639 --> 27636 --> 27636

The trajectory terminates when the number has a sum of primes and sum of non-primes that are equal, here 2 + 7 + 3 = 6 + 6. Numbers such as these are NOT listed in the OEIS but there is a listing for prime numbers with this property. There are 137 such primes in the range up to 40,000.

A371352: prime numbers such that the sum of their prime digits is equal to the sum of their nonprime digits.

The sequence begins (permalink):

167, 211, 541, 617, 761, 853, 1021, 1201, 1423, 1559, 1607, 1973, 2011, 2143, 2341, 2383, 2833, 3467, 3719, 3823, 3917, 4051, 4231, 4637, 4673, 5261, 5443, 5519, 5591, 6473, 6521, 6701, 7193, 7643, 7687, 7867, 8053, 8233, 8677, 9137, 9173, 9371, 9551, 10067, 10243, 10559, 10607, 10739, 10937, 10973, 11471, 11783, 12043, 12263, 12539, 12953, 13187, 13259, 13709, 13781, 13907, 14779, 14797, 15329, 15361, 15581, 15923, 16007, 16223, 17093, 17183, 17497, 17749, 17903, 18371, 18713, 18731, 19037, 19073, 19477, 20011, 20101, 20143, 20341, 20431, 21001, 22613, 23041, 23159, 24103, 25391, 25463, 25601, 25643, 25867, 25913, 25931, 26321, 26501, 28657, 29153, 29531, 30197, 30241, 30467, 30971, 31079, 31259, 31817, 31907, 32083, 32159, 32261, 32401, 32443, 32621, 32803, 33479, 33749, 34607, 34739, 35129, 35291, 36151, 37019, 37181, 37493, 37811, 37879, 37897, 37987, 38711, 38977, 39107, 39251, 39521, 39877

Many number trajectories will end in what I call an "attractor", using my Odds and Evens terminology. This is a number whose sums of prime and non-prime digits are equal. For completeness, here is a list of all the 446 NUMBERS (prime or composite) greater than 27630 and less than 40000 with the property that the sum of their prime digits equals the sum of their non-prime digits (permalink):

27636, 27658, 27663, 27685, 27788, 27801, 27810, 27834, 27843, 27856, 27865, 27878, 27887, 27900, 28017, 28033, 28071, 28107, 28125, 28152, 28170, 28215, 28222, 28251, 28303, 28330, 28347, 28374, 28437, 28455, 28473, 28512, 28521, 28545, 28554, 28567, 28576, 28657, 28675, 28701, 28710, 28734, 28743, 28756, 28765, 28778, 28787, 28877, 29007, 29025, 29052, 29070, 29135, 29153, 29205, 29223, 29232, 29250, 29315, 29322, 29351, 29502, 29513, 29520, 29531, 29700, 30036, 30058, 30063, 30085, 30111, 30124, 30142, 30179, 30197, 30214, 30238, 30241, 30283, 30306, 30328, 30339, 30360, 30382, 30393, 30412, 30421, 30445, 30454, 30467, 30476, 30508, 30544, 30580, 30603, 30630, 30647, 30674, 30719, 30746, 30764, 30791, 30805, 30823, 30832, 30850, 30917, 30933, 30971, 31011, 31024, 31042, 31079, 31097, 31101, 31110, 31134, 31143, 31156, 31165, 31178, 31187, 31204, 31226, 31240, 31259, 31262, 31295, 31314, 31338, 31341, 31383, 31402, 31413, 31420, 31431, 31516, 31529, 31561, 31592, 31615, 31622, 31651, 31709, 31718, 31781, 31790, 31817, 31833, 31871, 31907, 31925, 31952, 31970, 32014, 32038, 32041, 32083, 32104, 32126, 32140, 32159, 32162, 32195, 32216, 32229, 32261, 32292, 32308, 32344, 32380, 32401, 32410, 32434, 32443, 32456, 32465, 32478, 32487, 32519, 32546, 32564, 32591, 32612, 32621, 32645, 32654, 32667, 32676, 32748, 32766, 32784, 32803, 32830, 32847, 32874, 32915, 32922, 32951, 33006, 33028, 33039, 33060, 33082, 33093, 33114, 33138, 33141, 33183, 33208, 33244, 33280, 33309, 33318, 33381, 33390, 33411, 33424, 33442, 33479, 33497, 33600, 33749, 33794, 33802, 33813, 33820, 33831, 33903, 33930, 33947, 33974, 34012, 34021, 34045, 34054, 34067, 34076, 34102, 34113, 34120, 34131, 34201, 34210, 34234, 34243, 34256, 34265, 34278, 34287, 34311, 34324, 34342, 34379, 34397, 34405, 34423, 34432, 34450, 34504, 34526, 34540, 34559, 34562, 34595, 34607, 34625, 34652, 34670, 34706, 34728, 34739, 34760, 34782, 34793, 34827, 34872, 34937, 34955, 34973, 35008, 35044, 35080, 35116, 35129, 35161, 35192, 35219, 35246, 35264, 35291, 35404, 35426, 35440, 35459, 35462, 35495, 35549, 35594, 35611, 35624, 35642, 35679, 35697, 35769, 35796, 35800, 35912, 35921, 35945, 35954, 35967, 35976, 36003, 36030, 36047, 36074, 36115, 36122, 36151, 36212, 36221, 36245, 36254, 36267, 36276, 36300, 36407, 36425, 36452, 36470, 36511, 36524, 36542, 36579, 36597, 36627, 36672, 36704, 36726, 36740, 36759, 36762, 36795, 36957, 36975, 37019, 37046, 37064, 37091, 37109, 37118, 37181, 37190, 37248, 37266, 37284, 37349, 37394, 37406, 37428, 37439, 37460, 37482, 37493, 37569, 37596, 37604, 37626, 37640, 37659, 37662, 37695, 37789, 37798, 37811, 37824, 37842, 37879, 37897, 37901, 37910, 37934, 37943, 37956, 37965, 37978, 37987, 38005, 38023, 38032, 38050, 38117, 38133, 38171, 38203, 38230, 38247, 38274, 38302, 38313, 38320, 38331, 38427, 38472, 38500, 38711, 38724, 38742, 38779, 38797, 38977, 39017, 39033, 39071, 39107, 39125, 39152, 39170, 39215, 39222, 39251, 39303, 39330, 39347, 39374, 39437, 39455, 39473, 39512, 39521, 39545, 39554, 39567, 39576, 39657, 39675, 39701, 39710, 39734, 39743, 39756, 39765, 39778, 39787, 39877

Also for completeness, here is a list of all the numbers greater than 27630 and less than 40000 with the property they map back to themselves after two or more repetitions of the prime / non-prime algorithm. In other words, they are "vorticals" each belonging to a "vortex" to use my Odds and Evens nomenclature.

27675, 27684, 27690, 27823, 27829, 27837, 27848, 27855, 27866, 27873, 27884, 27922, 27926, 27933, 27939, 27954, 27955, 27964, 27965, 27972, 27981, 28027, 28030, 28272, 28273, 28277, 28279, 28282, 28287, 28352, 28355, 28356, 28362, 28532, 28536, 28553, 28560, 28723, 28729, 28737, 28748, 28755, 28766, 28773, 28784, 29272, 29276, 29372, 29373, 29377, 29379, 29382, 29387, 29507, 29511, 29512, 29552, 29553, 29557, 29559, 29566, 29567, 29570, 29575, 29580, 29585, 29722, 29726, 29733, 29739, 29754, 29755, 29764, 29765, 29772, 29781, 30003, 30009, 30012, 30016, 30075, 30084, 30090, 30102, 30106, 30135, 30138, 30145, 30148, 30259, 30260, 30277, 30286, 30296, 30342, 30346, 30378, 30383, 30384, 30387, 30391, 30392, 30432, 30436, 30453, 30460, 30562, 30566, 30652, 30655, 30656, 30662, 30782, 30786, 30795, 30801, 30872, 30876, 30927, 30930, 31002, 31006, 31035, 31038, 31045, 31048, 31123, 31129, 31137, 31148, 31155, 31166, 31173, 31184, 31213, 31219, 31276, 31281, 31359, 31360, 31377, 31386, 31396, 31407, 31411, 31412, 31452, 31453, 31457, 31459, 31466, 31467, 31470, 31475, 31480, 31485, 31543, 31549, 31579, 31584, 31672, 31673, 31677, 31679, 31682, 31687, 31763, 31769, 31827, 31830, 32059, 32060, 32077, 32086, 32096, 32113, 32119, 32176, 32181, 32243, 32249, 32279, 32284, 32362, 32366, 32423, 32429, 32437, 32448, 32455, 32466, 32473, 32484, 32582, 32586, 32632, 32636, 32653, 32660, 32793, 32797, 32799, 32806, 32807, 32811, 32852, 32855, 32856, 32862, 32972, 32973, 32977, 32979, 32982, 32987, 33042, 33046, 33078, 33083, 33084, 33087, 33091, 33092, 33159, 33160, 33177, 33186, 33196, 33262, 33266, 33363, 33369, 33402, 33406, 33435, 33438, 33445, 33448, 33583, 33589, 33592, 33596, 33622, 33626, 33633, 33639, 33654, 33655, 33664, 33665, 33672, 33681, 33807, 33811, 33812, 33852, 33853, 33857, 33859, 33866, 33867, 33870, 33875, 33880, 33885, 33952, 33955, 33956, 33962, 34032, 34036, 34053, 34060, 34107, 34111, 34112, 34152, 34153, 34157, 34159, 34166, 34167, 34170, 34175, 34180, 34185, 34223, 34229, 34237, 34248, 34255, 34266, 34273, 34284, 34302, 34306, 34335, 34338, 34345, 34348, 34472, 34476, 34513, 34519, 34576, 34581, 34742, 34746, 34778, 34783, 34784, 34787, 34791, 34792, 35062, 35066, 35143, 35149, 35179, 35184, 35282, 35286, 35383, 35389, 35392, 35396, 35413, 35419, 35476, 35481, 35602, 35606, 35635, 35638, 35645, 35648, 35822, 35826, 35833, 35839, 35854, 35855, 35864, 35865, 35872, 35881, 35932, 35936, 35953, 35960, 36052, 36055, 36056, 36062, 36172, 36173, 36177, 36179, 36182, 36187, 36232, 36236, 36253, 36260, 36322, 36326, 36333, 36339, 36354, 36355, 36364, 36365, 36372, 36381, 36502, 36506, 36535, 36538, 36545, 36548, 36713, 36719, 36776, 36781, 37082, 37086, 37163, 37169, 37293, 37299, 37442, 37446, 37478, 37483, 37484, 37487, 37491, 37492, 37613, 37619, 37676, 37681, 37802, 37806, 37835, 37838, 37845, 37848, 37923, 37929, 37937, 37948, 37955, 37966, 37973, 37984, 38072, 38076, 38127, 38130, 38252, 38255, 38256, 38262, 38307, 38311, 38312, 38352, 38353, 38357, 38359, 38366, 38367, 38370, 38375, 38380, 38385, 38522, 38526, 38533, 38539, 38554, 38555, 38564, 38565, 38572, 38581, 38702, 38706, 38735, 38738, 38745, 38748, 39027, 39030, 39272, 39273, 39277, 39279, 39282, 39287, 39352, 39355, 39356, 39362, 39532, 39536, 39553, 39560, 39723, 39729, 39737, 39748, 39755, 39766, 39773, 39784

Take for example, the first member of the above series of numbers:

27675 --> 27690 --> 27684 --> 27675

Here (27675, 27690, 27684) forms a vortex while the individual members (27675, 27690 and 27684) are vorticals. If a number is not an attractor or a vortical then it will captured either by an attractor or a vortex. For example, 27676 is captured by the vortex just mentioned:

27676 --> 27680 --> 27675 --> 27690 --> 27684 --> 27675

On the other hand, 27659 is captured by the attractor 27658:

27659 --> 27658 --> 27658

I've now included in my multipurpose algorithm the trajectory of a number under the prime and non-prime algorithm. 

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.

Attractors, Vortices and Captives

 Regarding my previous post, I'm inclining toward the following nomenclature:

• an attractor to describe a number that is invariant under the odd-even recursive process
  
  
• a vortex to describe an loop involving two or more numbers under the process
  
  
• a vortical to describe a number that forms part of a vortex
  
  
• a captive to describe a number that eventually leads to an attractor or a vortex
  
  
• N-captive to describe a number that is captive to a number N that is either an attractor or the smallest member of a vortex

The trajectory of 710 leads directly to the attractor 718 so i710 can be described as a 718-captive. The trajectory of 719 leads to 719, 736, 740, 743, 749, 761, 763, 767, 775, 794, 806, 792, 806 and is thus captured by the 792-806 vortex. As a convention, I'll choose the smallest vortical to identify the vortex (although any vortical would do) and so 719 becomes a 792-captive. 

It's clear from the leading number whether the number to captive to an attractor or a vortex but a subscript could be optionally added for numbers with a large number of digits. Thus we could write 718\( _a \)-captive and 782\( _v \)-captive.

Attractors can be visualised as having a solid central core (the attractor itself) with various spokes corresponding to the captives attached to it. For example, 718 is an attractor with four captives: 710, 712, 714 and 716. Each of these four captives is only one step removed from the attractor. This can be represented as shown in Figure 1.

Figure 1: 718 is an attractor with four captives

Some attractors have no captives. For example, in the range from 690 to 889, there are only six attractors: 718, 781, 817, 835, 853 and 871. Of these, only 718 has any captives. On the other hand, some attractors like 87980 with 881 captives can be termed great attractors.

Meanwhile, in the aforementioned range from 690 to 889, every number (apart from the six attractors and four captives) is captive to the small but powerful vortex: 792-806. These two numbers are vorticals and together make up the vortex. Such a vortex could be represented as shown in Figure 2.

Figure 2: the vortex 792-806

A vortex can be shown with its captives attached, although there are too many to show in the case of 792-806. All captives shown can be described as 792-captives. See Figure 3.

Figure 3: vortex, vorticals and captives

Attractors in general could be represented by A and would be equivalent to the number itself and could be differentiated by their subscripts \(A_1, A_2, \dots \). Thus we could write \(A_1=\left \{718 \right \} \). An attractor can also be associated with the set of its captives. For example, an attractor \(A_1\) with \(n\) captives could be associated with the set \(C_1\) such that:$$C_1=\left \{c_1, c_1, \dots , c_{n-1}, c_n \right \}$$ A particular example is \(C_1= \left \{710, 712, 714, 716 \right \} \) where \(A_1= \left \{ 718 \right \} \).

A vortex could be represented by V and would be equivalent to the set of its vorticals v. Thus for a particular vortex \(V_1\), with \(n\) vorticals, could be written as:$$V_1=\left \{ v_1, v_2, \dots , v_{n-1},v_n \right \}$$A particular example would be \(V_1=\left \{792, 806 \right \} \).

A vortex is also associated with the set of its captives and so a particular vortex \(V_1\) with \(n\) captives could be associated with the set \(C_2\) such that:$$C_2=\left \{c_1, c_1, \dots , c_{n-1}, c_n \right \}$$That's about it for this post. I just wanted to establish a consistent and readily understandable notational system. I'm just developing this as I go so there may well be future modifications.