PRIME + and NON-PRIME - Improved Format

As with the EVEN + and ODD - and EVEN - and ODD + recursive algorithms, I've taken the original 393 page data document and converted it to a 83 page PDF that is more reader friendly. Here some excerpts.

Figure 1

Figure 2

Figure 3

Table 1

Table 2

Table 3

Table 4

ODD + and EVEN - Improved Format

Some time ago I got Gemini to carry out an analysis for me of attractors and vortices in the range up to 40000 under the ODD + and EVEN - algorithm. I copied the output into a Google document that listed: 

• each attractor and how many captives it had
• each vortex (along with the vorticals that comprised it)
• the number of captives the vortex had

Here is the information the 193 page document displays for the attractor 39642:

• Attractor: 39642
• Captive Count: 395
• Captives: [38997, 39017, 39035, 39037, 39039, 39053, 39055, 39059, 39063, 39065, 39070, 39071, 39072, 39073, 39074, 39075, 39076, 39077, 39078, 39083] ... (and 375 more)

Here is the information the documents displays for the vortex [38013, 38012, 38006, 37995, 38028]:

• Vortex (Vorticals): [38013, 38012, 38006, 37995, 38028]
• Captive Count: 564
• Captives: [37055, 37075, 37097, 37099, 37107, 37123, 37125, 37127, 37133, 37135, 37137, 37139, 37141, 37143, 37145, 37149, 37150, 37151, 37152, 37153] ... (and 544 more)

The attractors and vortices are arranged in descending order by number of captives. The information is readily accessible but I thought I'd use Gemini again to improve the formatting of the document to make it more readable. Figure 1 shows the opening page of the 137 page document and Figure 2 shows the second page.

Figure 1

Figure 2

The document lists the attractors in ascending order and also in descending order by number of captives (see Tables 1 and 2 for the start of each table).

Table 1

Table 2

The document goes on to list vortices in ascending order and also in descending order by number of captives (see Tables 3 and 4):

Table 3

Table 4

Overall the document provides an excellent organisation of the data and I'll roll this out this the results for the ODD - and EVEN + algorithm and the PRIME + NON-PRIME - algorithm in the near future.

Code for Attractors, Vortices and Captives

Herein is an attempt to organise the code that I've gotten Gemini to write for me regarding attractors, vortices and captives.

Firstly, let's start with the ODD(+) and EVEN(-) algorithm. Here is a permalink to the code that will generate a list of attractors and vortices in decreasing order of their number of captives. It will also generate a summary and a colour-coded graphical display (see Table 1 and Figure 1). The default range is 0 to 40000.

Table 1: ODD(+) and EVEN(-)

Figure 1: red = attractor, orange = vortex, blue = captive

Secondly, let's continue with the ODD(-) and EVEN(+) algorithm. Here is a permalink to the code that will generate a list of attractors and vortices in decreasing order of their number of captives. It will also generate a summary and a colour-coded graphical display (see Table 2 and Figure 2). The default range is 0 to 40000.

Table 2: ODD(-) and EVEN(+)

Figure 2: red = attractor, orange = vortex, blue = captive

Thirdly, let's continue with the PRIME(+) and NON-PRIME(-) algorithm. Here is a permalink to the code that will generate a list of attractors and vortices in decreasing order of their number of captives. It will also generate a summary and a colour-coded graphical display (see Table 3 and Figure 3). The default range is 0 to 40000.

Table 3: PRIME(+) and NON-PRIME(-)

Figure 3: red = attractor, orange = vortex, blue = captive

Fourthly, let's continue with the PRIME(-) and NON-PRIME(+) algorithm. Here is a permalink to the code that will generate a list of attractors and vortices in decreasing order of their number of captives. It will also generate a summary and a colour-coded graphical display (see Table 4 and Figure 4). The default range is 0 to 40000.

Table 4: PRIME(-) and NON-PRIME(+)

Figure 4:  red = attractor, orange = vortex, blue = captive

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

Revisiting Odds And Evens

It's been a while since I posted about so-called odds and evens by which I mean the process of generating a new number from an existing one by adding its sum of odd digits and subtracting it's sum of even digits. An an example, let's consider 1234:$$1234 \rightarrow 1234+1+3-2-4=1232$$As we continue this process interesting things happen and that's what I wrote about in a paper that I published to Academia (link). Here are links to posts I've made about the topic:

• 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

Of course, if the sums of the odd and even digits are the same, the number remains unchanged. These sorts of numbers are what I termed "attractors" because other numbers, wherein there is an imbalance of odds and evens, have either these as their termini or they enter "vortices" or loops. My diurnal age today, 27326, is one such attractor which is why I was reminded of them. $$27326 \rightarrow 27326 + 7 + 3-2-2-6=27326$$Until now, I wasn't aware that they had their own OEIS sequence but they do and it is OEIS A036301 (permalink):

A036301

Numbers whose sum of even digits and sum of odd digits are equal.

The initial members of the sequence are:

0, 112, 121, 134, 143, 156, 165, 178, 187, 211, 314, 336, 341, 358, 363, 385, 413, 431, 516, 538, 561, 583, 615, 633, 651, 718, 781, 817, 835, 853, 871, 1012, 1021, 1034, 1043, 1056, 1065, 1078, 1087, 1102, 1120, 1201, 1210, 1223, 1232, 1245, 1254, 1267, 1276, 1289, 1298

I also discovered a sequence connected to what I termed "vorticals". These are the numbers that comprise a vortex into which non-balanced numbers are sucked if they do not end up in an attractor. The first instance of a vortex is that involving the numbers 11 and 13:$$ \begin{align} 11 \rightarrow 11 + 1 + 1 &= 13\\13 \rightarrow 13 + 1 + 3 &=17\\ 17 \rightarrow 17+1+7 &=25\\25 \rightarrow 25+5-2 &=28\\28 \rightarrow 28-2-8 &=18\\18 \rightarrow 18+1 -8 &=11 \end{align}$$Thus 11, 13, 17, 18, 25 and 28 are all numbers that eventually return to themselves after repeated mappings involving the odds and evens recursion. In the case of these numbers, six repetitions are needed: $$11 \rightarrow 13 \rightarrow 17 \rightarrow 25 \rightarrow 28 \rightarrow 18 \rightarrow 11$$Numbers like these are included in OEIS A124176:

A124176

Consider the map \(f\) that sends \(m\) to \(m\) + (sum of odd digits of \(m\)) - (sum of even digits of \(m\)). Sequence gives numbers \(m\) such that \(f^k(m) \) = \(m\) for some \(k\).

This will include all the attractors, the numbers that aren't changed by the mapping and where \(k\)=1. The initial members are (permalink):

0, 11, 13, 17, 18, 25, 28, 54, 55, 64, 65, 112, 121, 134, 137, 143, 148, 155, 156, 165, 166, 173, 178, 184, 187, 198, 200, 209, 211, 216, 231, 233, 234, 237, 244, 245, 270, 275, 280, 285, 314, 336, 341, 358, 363, 385, 396, 402, 407, 410, 413, 429, 431, 432

All other numbers are what I termed "captives" because they are "captured" by either an attractor or a "vortex" that is comprised of "vorticals". The attractors and vorticals are all listed in OEIS A124176 and OEIS A036301 is a subset of this sequence because it contains only the attractors that are invariant under the mapping.

There are 2974 such numbers in the range up to 40,000. Here are the numbers from 27326 up to 40000 that belong in the sequence (for future reference):

27326, 27344, 27362, 27380, 27412, 27421, 27434, 27437, 27443, 27448, 27455, 27456, 27465, 27466, 27473, 27478, 27484, 27487, 27528, 27546, 27564, 27582, 27601, 27610, 27617, 27623, 27624, 27632, 27635, 27642, 27645, 27653, 27654, 27660, 27667, 27676, 27689, 27698, 27748, 27766, 27784, 27803, 27825, 27830, 27847, 27852, 27869, 27874, 27896, 27968, 27986, 27990, 28004, 28007, 28013, 28019, 28037, 28055, 28073, 28091, 28093, 28095, 28097, 28099, 28101, 28103, 28105, 28107, 28109, 28190, 28239, 28257, 28275, 28293, 28295, 28297, 28299, 28301, 28303, 28305, 28307, 28329, 28370, 28392, 28459, 28477, 28495, 28497, 28499, 28501, 28503, 28505, 28527, 28549, 28550, 28572, 28594, 28679, 28697, 28699, 28701, 28703, 28725, 28730, 28747, 28752, 28769, 28774, 28796, 28899, 28901, 28910, 28917, 28923, 28924, 28932, 28935, 28942, 28945, 28953, 28954, 28960, 28967, 28976, 28989, 28998, 29018, 29081, 29108, 29126, 29144, 29162, 29180, 29216, 29238, 29261, 29283, 29328, 29346, 29364, 29382, 29414, 29436, 29441, 29458, 29463, 29485, 29548, 29566, 29584, 29612, 29621, 29634, 29637, 29643, 29648, 29655, 29656, 29665, 29666, 29673, 29678, 29684, 29687, 29768, 29786, 29801, 29810, 29817, 29823, 29824, 29832, 29835, 29842, 29845, 29853, 29854, 29860, 29867, 29876, 29889, 29898, 29988, 30014, 30036, 30041, 30058, 30063, 30085, 30104, 30122, 30140, 30178, 30181, 30212, 30221, 30234, 30237, 30243, 30248, 30255, 30256, 30265, 30266, 30273, 30278, 30284, 30287, 30306, 30324, 30342, 30360, 30378, 30383, 30384, 30401, 30410, 30417, 30423, 30424, 30432, 30435, 30442, 30445, 30453, 30454, 30460, 30467, 30476, 30489, 30498, 30508, 30526, 30544, 30562, 30580, 30597, 30603, 30608, 30615, 30617, 30618, 30621, 30622, 30625, 30630, 30647, 30652, 30669, 30674, 30696, 30728, 30746, 30764, 30782, 30795, 30805, 30806, 30812, 30813, 30819, 30824, 30827, 30849, 30850, 30872, 30894, 30948, 30966, 30984, 31004, 31022, 31040, 31078, 31081, 31116, 31138, 31161, 31183, 31202, 31220, 31235, 31238, 31245, 31248, 31318, 31381, 31400, 31411, 31413, 31417, 31418, 31425, 31428, 31454, 31455, 31464, 31465, 31598, 31608, 31609, 31611, 31616, 31631, 31633, 31634, 31637, 31644, 31645, 31670, 31675, 31680, 31685, 31807, 31810, 31813, 31829, 31831, 31832, 31850, 31851, 31853, 31857, 31860, 31865, 32012, 32021, 32034, 32037, 32043, 32048, 32055, 32056, 32065, 32066, 32073, 32078, 32084, 32087, 32102, 32120, 32135, 32138, 32145, 32148, 32201, 32210, 32217, 32223, 32224, 32232, 32235, 32242, 32245, 32253, 32254, 32260, 32267, 32276, 32289, 32298, 32304, 32322, 32340, 32378, 32381, 32397, 32403, 32408, 32415, 32417, 32418, 32422, 32425, 32430, 32447, 32452, 32469, 32474, 32496, 32506, 32524, 32542, 32560, 32578, 32583, 32584, 32595, 32605, 32606, 32615, 32616, 32627, 32649, 32650, 32672, 32694, 32708, 32726, 32744, 32762, 32780, 32793, 32804, 32807, 32810, 32813, 32829, 32870, 32892, 32928, 32946, 32964, 32982, 33006, 33024, 33042, 33060, 33078, 33083, 33084, 33118, 33181, 33204, 33222, 33240, 33278, 33281, 33402, 33420, 33435, 33438, 33445, 33448, 33600, 33611, 33613, 33617, 33618, 33625, 33628, 33654, 33655, 33664, 33665, 33809, 33811, 33816, 33831, 33833, 33834, 33837, 33844, 33845, 33870, 33875, 33880, 33885, 34001, 34010, 34017, 34023, 34024, 34032, 34035, 34042, 34045, 34053, 34054, 34060, 34067, 34076, 34089, 34098, 34100, 34111, 34113, 34117, 34118, 34125, 34128, 34154, 34155, 34164, 34165, 34197, 34203, 34208, 34213, 34214, 34225, 34230, 34247, 34252, 34269, 34274, 34296, 34302, 34320, 34335, 34338, 34345, 34348, 34395, 34405, 34408, 34411, 34427, 34449, 34450, 34472, 34494, 34504, 34522, 34540, 34578, 34581, 34593, 34606, 34607, 34609, 34611, 34629, 34670, 34692, 34706, 34724, 34742, 34760, 34778, 34783, 34784, 34791, 34793, 34801, 34804, 34805, 34807, 34809, 34811, 34890, 34908, 34926, 34944, 34962, 34980, 35008, 35026, 35044, 35062, 35080, 35206, 35224, 35242, 35260, 35278, 35283, 35284, 35404, 35422, 35440, 35478, 35481, 35602, 35620, 35635, 35638, 35645, 35648, 35800, 35811, 35813, 35817, 35818, 35825, 35828, 35854, 35855, 35864, 35865, 36003, 36025, 36030, 36047, 36052, 36069, 36074, 36096, 36109, 36111, 36116, 36131, 36133, 36134, 36137, 36144, 36145, 36170, 36175, 36180, 36185, 36195, 36205, 36206, 36207, 36209, 36212, 36213, 36227, 36249, 36250, 36272, 36294, 36300, 36311, 36313, 36317, 36318, 36325, 36328, 36354, 36355, 36364, 36365, 36393, 36399, 36403, 36404, 36405, 36407, 36417, 36418, 36429, 36470, 36492, 36502, 36520, 36535, 36538, 36545, 36548, 36591, 36593, 36597, 36601, 36602, 36603, 36605, 36607, 36609, 36612, 36615, 36690, 36704, 36722, 36740, 36778, 36781, 36789, 36794, 36795, 36803, 36806, 36813, 36906, 36924, 36942, 36960, 36978, 36983, 36984, 37028, 37046, 37064, 37082, 37208, 37226, 37244, 37262, 37280, 37406, 37424, 37442, 37460, 37478, 37483, 37484, 37604, 37622, 37640, 37678, 37681, 37802, 37820, 37835, 37838, 37845, 37848, 37995, 38005, 38006, 38012, 38013, 38027, 38028, 38049, 38050, 38072, 38094, 38107, 38110, 38113, 38129, 38131, 38132, 38150, 38151, 38153, 38157, 38160, 38165, 38193, 38195, 38199, 38201, 38203, 38204, 38205, 38207, 38210, 38213, 38229, 38270, 38292, 38309, 38311, 38316, 38331, 38333, 38334, 38337, 38344, 38345, 38370, 38375, 38380, 38385, 38391, 38399, 38402, 38409, 38412, 38415, 38490, 38500, 38511, 38513, 38517, 38518, 38525, 38528, 38554, 38555, 38564, 38565, 38589, 38590, 38591, 38595, 38599, 38600, 38601, 38603, 38607, 38609, 38614, 38617, 38702, 38720, 38735, 38738, 38745, 38748, 38774, 38779, 38787, 38788, 38789, 38792, 38797, 38801, 38808, 38815, 38904, 38922, 38940, 38978, 38981, 39048, 39066, 39084, 39228, 39246, 39264, 39282, 39408, 39426, 39444, 39462, 39480, 39606, 39624, 39642, 39660, 39678, 39683, 39684, 39804, 39822, 39840, 39878, 39881

Of course, the choice of adding the odd digits and subtracting the even digits is quite arbitrary and it's perfectly acceptable to reverse this and in doing so generate a different sequence. Doing this we generate OEIS A124177:

A124177

Consider the map \(f\) that sends \(m\) to \(m\) + (sum of even digits of \(m\)) - (sum of odd digits of \(m\) ). Sequence gives numbers \(m\) such that \(f^k(m)\) = \(m\) for some \(k\).

The initial members of this sequence are:

0, 22, 26, 27, 34, 35, 44, 49, 52, 63, 66, 78, 79, 81, 88, 99, 104, 107, 108, 112, 115, 121, 126, 133, 134, 143, 144, 151, 156, 165, 178, 187, 211, 224, 229, 232, 233, 283, 290, 314, 336, 341, 358, 363, 385, 413, 431, 467, 470, 489, 492, 516, 538, 561, 583, 615

Let's look at the first member of the sequence, 22:$$ \begin{align} 22 \rightarrow 22 + 2 + 2 &=26\\26 \rightarrow 26+2+6 &=34\\34 \rightarrow 34 + 4 - 3 &= 35\\35 \rightarrow 35-3-5 &=27\\27 \rightarrow 27+2-7 &=22 \end{align}$$Thus after five steps we arrive back at 22:$$22 \rightarrow 26 \rightarrow 34 \rightarrow 35 \rightarrow 27 \rightarrow 22$$It can be noted that "attractors", those numbers invariant under the mapping, remain the same but the "vorticals", those numbers that form "vortices" or loops, are different. There are 2966 members of this sequence in the range up to 40,000. For future reference, here are the members from 27326 to 40000:

27326, 27344, 27362, 27380, 27408, 27412, 27415, 27421, 27426, 27433, 27434, 27443, 27444, 27451, 27456, 27465, 27478, 27487, 27528, 27546, 27564, 27582, 27601, 27610, 27623, 27628, 27632, 27639, 27645, 27646, 27654, 27657, 27664, 27667, 27675, 27676, 27682, 27688, 27689, 27693, 27698, 27705, 27748, 27766, 27784, 27803, 27825, 27830, 27847, 27852, 27869, 27874, 27884, 27886, 27896, 27899, 27903, 27968, 27986, 28019, 28037, 28055, 28073, 28091, 28109, 28190, 28239, 28257, 28275, 28293, 28307, 28329, 28370, 28392, 28459, 28477, 28495, 28505, 28527, 28549, 28550, 28572, 28594, 28679, 28697, 28703, 28725, 28730, 28747, 28752, 28769, 28774, 28784, 28796, 28799, 28899, 28901, 28910, 28923, 28928, 28932, 28939, 28945, 28946, 28954, 28957, 28964, 28967, 28975, 28976, 28982, 28989, 28993, 28998, 29018, 29081, 29108, 29126, 29144, 29162, 29180, 29216, 29238, 29261, 29283, 29328, 29346, 29364, 29382, 29414, 29436, 29441, 29458, 29463, 29485, 29488, 29501, 29548, 29566, 29584, 29608, 29612, 29615, 29621, 29626, 29633, 29634, 29643, 29644, 29651, 29656, 29665, 29678, 29687, 29768, 29786, 29801, 29810, 29823, 29828, 29832, 29839, 29845, 29846, 29854, 29857, 29864, 29867, 29875, 29876, 29882, 29889, 29893, 29898, 29988, 30014, 30036, 30041, 30058, 30063, 30085, 30104, 30118, 30121, 30122, 30140, 30162, 30166, 30167, 30174, 30181, 30184, 30192, 30208, 30212, 30215, 30221, 30226, 30233, 30234, 30243, 30244, 30251, 30256, 30265, 30278, 30287, 30288, 30294, 30303, 30306, 30324, 30342, 30360, 30382, 30383, 30386, 30394, 30401, 30410, 30423, 30428, 30432, 30439, 30445, 30446, 30454, 30457, 30464, 30467, 30475, 30476, 30482, 30486, 30489, 30492, 30493, 30498, 30501, 30508, 30526, 30544, 30562, 30580, 30603, 30625, 30630, 30647, 30652, 30669, 30674, 30684, 30696, 30699, 30728, 30746, 30764, 30782, 30805, 30827, 30849, 30850, 30872, 30888, 30894, 30909, 30948, 30966, 30984, 31004, 31018, 31021, 31022, 31040, 31062, 31066, 31067, 31074, 31081, 31084, 31092, 31116, 31138, 31161, 31183, 31202, 31220, 31242, 31243, 31246, 31251, 31254, 31264, 31265, 31272, 31318, 31381, 31400, 31422, 31426, 31427, 31434, 31435, 31444, 31449, 31452, 31463, 31466, 31478, 31479, 31481, 31488, 31499, 31504, 31611, 31624, 31629, 31632, 31633, 31683, 31690, 31813, 31831, 31867, 31870, 31889, 31892, 32008, 32012, 32015, 32021, 32026, 32033, 32034, 32043, 32044, 32051, 32056, 32065, 32078, 32087, 32102, 32120, 32142, 32143, 32146, 32151, 32154, 32164, 32165, 32172, 32201, 32210, 32223, 32228, 32232, 32239, 32245, 32246, 32254, 32257, 32264, 32267, 32275, 32276, 32282, 32289, 32293, 32298, 32304, 32318, 32321, 32322, 32340, 32362, 32366, 32367, 32374, 32381, 32384, 32392, 32403, 32425, 32430, 32447, 32452, 32469, 32474, 32484, 32488, 32492, 32494, 32496, 32499, 32506, 32507, 32524, 32542, 32560, 32582, 32583, 32586, 32594, 32605, 32627, 32649, 32650, 32672, 32686, 32690, 32692, 32694, 32696, 32698, 32702, 32705, 32708, 32726, 32744, 32762, 32780, 32807, 32829, 32870, 32888, 32892, 32898, 32899, 32904, 32911, 32928, 32946, 32964, 32982, 33006, 33024, 33042, 33060, 33082, 33083, 33086, 33094, 33118, 33181, 33204, 33218, 33221, 33222, 33240, 33262, 33266, 33267, 33274, 33281, 33284, 33292, 33402, 33420, 33442, 33443, 33446, 33451, 33454, 33464, 33465, 33472, 33600, 33622, 33626, 33627, 33634, 33635, 33644, 33649, 33652, 33663, 33666, 33678, 33679, 33681, 33688, 33695, 33704, 33811, 33824, 33829, 33832, 33833, 33883, 33890, 34001, 34010, 34023, 34028, 34032, 34039, 34045, 34046, 34054, 34057, 34064, 34067, 34075, 34076, 34082, 34089, 34093, 34098, 34100, 34122, 34126, 34127, 34134, 34135, 34144, 34149, 34152, 34163, 34166, 34178, 34179, 34203, 34225, 34230, 34247, 34252, 34269, 34274, 34284, 34296, 34298, 34299, 34300, 34302, 34320, 34342, 34343, 34346, 34351, 34354, 34364, 34365, 34372, 34405, 34427, 34449, 34450, 34472, 34494, 34496, 34498, 34500, 34502, 34504, 34518, 34521, 34522, 34540, 34562, 34566, 34567, 34574, 34581, 34584, 34592, 34607, 34629, 34670, 34692, 34694, 34696, 34698, 34700, 34702, 34704, 34706, 34724, 34742, 34760, 34782, 34783, 34786, 34794, 34809, 34890, 34892, 34894, 34896, 34898, 34900, 34902, 34904, 34906, 34908, 34926, 34944, 34962, 34980, 35008, 35026, 35044, 35062, 35080, 35206, 35224, 35242, 35260, 35282, 35283, 35286, 35294, 35404, 35418, 35421, 35422, 35440, 35462, 35466, 35467, 35474, 35481, 35484, 35492, 35602, 35620, 35642, 35643, 35646, 35651, 35654, 35664, 35665, 35672, 35800, 35822, 35826, 35827, 35834, 35835, 35844, 35849, 35852, 35863, 35866, 35878, 35879, 35881, 35888, 35891, 35904, 36003, 36025, 36030, 36047, 36052, 36069, 36074, 36084, 36096, 36099, 36111, 36124, 36129, 36132, 36133, 36183, 36190, 36205, 36227, 36249, 36250, 36272, 36294, 36300, 36322, 36326, 36327, 36334, 36335, 36344, 36349, 36352, 36363, 36366, 36378, 36379, 36407, 36429, 36470, 36492, 36498, 36502, 36504, 36506, 36507, 36510, 36520, 36542, 36543, 36546, 36551, 36554, 36564, 36565, 36572, 36609, 36690, 36696, 36700, 36702, 36704, 36718, 36721, 36722, 36740, 36762, 36766, 36767, 36774, 36781, 36784, 36792, 36894, 36900, 36906, 36924, 36942, 36960, 36982, 36983, 36986, 36994, 37028, 37046, 37064, 37082, 37208, 37226, 37244, 37262, 37280, 37406, 37424, 37442, 37460, 37482, 37483, 37486, 37494, 37604, 37618, 37621, 37622, 37640, 37662, 37666, 37667, 37674, 37681, 37684, 37692, 37802, 37820, 37842, 37843, 37846, 37851, 37854, 37864, 37865, 37872, 38005, 38027, 38049, 38050, 38072, 38094, 38113, 38131, 38167, 38170, 38189, 38192, 38207, 38229, 38270, 38292, 38311, 38324, 38329, 38332, 38333, 38383, 38390, 38409, 38490, 38500, 38522, 38526, 38527, 38534, 38535, 38544, 38549, 38552, 38563, 38566, 38578, 38579, 38698, 38702, 38707, 38708, 38714, 38715, 38720, 38742, 38743, 38746, 38751, 38754, 38764, 38765, 38772, 38896, 38904, 38906, 38908, 38909, 38912, 38918, 38921, 38922, 38940, 38962, 38966, 38967, 38974, 38981, 38984, 38988, 38992, 39000, 39048, 39066, 39084, 39228, 39246, 39264, 39282, 39408, 39426, 39444, 39462, 39480, 39606, 39624, 39642, 39660, 39682, 39683, 39686, 39694, 39804, 39818, 39821, 39822, 39840, 39862, 39866, 39867, 39874, 39881, 39884, 39892

Second Order Odds and Evens Trajectory for Numbers 1 to 99

In my previous post titled Higher Order Odds and Evens Trajectory, I looked specifically at the trajectory of the number 26638 under the recursive rule that:

number --> number + \( \sum d_o^k - \sum d_e^k \) with \(k \geq 1\)

where \( d_o^k \) are the number's odd digits raised to the power \( k\) and \( d_e^k \) are the number's even digits raised to the power \( k\).  In that post, I looked at the behaviour for \(2 \leq k \leq 5 \). I've looked at the case of \(k=1\) for a wide variety of numbers in several posts back in 2021 so in this post I'm focusing on values of \(k=2\) and looking only at the numbers from 1 to 99. Thus the recursive rule here is:

number --> number + \( \sum d_o^2 - \sum d_e^2 \) 

The trajectory of 1 when \(k=2\) requires 35 steps to reach the loop {327, 381}, acquiring a maximum value of 428 in the process. See Figure 1.

Figure 1

The full details of the trajectory of 1 are as follows:

1, 2, -2, -6, -42, -62, -102, -105, -79, 51, 77, 175, 250, 271, 317, 376, 398, 424, 388, 269, 310, 320, 325, 355, 414, 383, 337, 404, 372, 426, 370, 428, 344, 321, 327, 381, 327

The trajectory of 2 is almost identical to that of 1 after only one step (shown in blue):

2, -2, -6, -42, -62, -102, -105, -79, 51, 77, 175, 250, 271, 317, 376, 398, 424, 388, 269, 310, 320, 325, 355, 414, 383, 337, 404, 372, 426, 370, 428, 344, 321, 327, 381, 327

The trajectory of 3 eventually overlaps the trajectory of 1 and 2 (shown in blue):

3, 12, 9, 90, 171, 222, 210, 207, 252, 269, 310, 320, 325, 355, 414, 383, 337, 404, 372, 426, 370, 428, 344, 321, 327, 381, 327

The trajectory of 4 also overlaps the trajectory of 1 (shown in blue) and is 49 steps in length:

4, -12, -15, 11, 13, 23, 28, -40, -56, -67, -54, -45, -36, -63, -90, -9, 72, 117, 168, 69, 114, 100, 101, 103, 113, 124, 105, 131, 142, 123, 129, 207, 252, 269, 310, 320, 325, 355, 414, 383, 337, 404, 372, 426, 370, 428, 344, 321, 327, 381, 327

The trajectory of 5 is quite short, at 22 steps, and it too overlaps the trajectory of 1 (shown in blue):

5, 30, 39, 129, 207, 252, 269, 310, 320, 325, 355, 414, 383, 337, 404, 372, 426, 370, 428, 344, 321, 327, 381, 327

The trajectory of 6 is 42 steps long and also overlaps the trajectory of 1 (shown in blue):

6, -30, -21, -24, -44, -76, -63, -90, -9, 72, 117, 168, 69, 114, 100, 101, 103, 113, 124, 105, 131, 142, 123, 129, 207, 252, 269, 310, 320, 325, 355, 414, 383, 337, 404, 372, 426, 370, 428, 344, 321, 327, 381, 327

The trajectory of 7 is 29 steps in length and overlaps the trajectory of 1 (shown in blue):

7, 56, 45, 54, 63, 36, 9, 90, 171, 222, 210, 207, 252, 269, 310, 320, 325, 355, 414, 383, 337, 404, 372, 426, 370, 428, 344, 321, 327, 381, 327

The trajectory of 8 is 42 steps in length and overlaps the trajectory of 1 (shown in blue):

8, -56, -67, -54, -45, -36, -63, -90, -9, 72, 117, 168, 69, 114, 100, 101, 103, 113, 124, 105, 131, 142, 123, 129, 207, 252, 269, 310, 320, 325, 355, 414, 383, 337, 404, 372, 426, 370, 428, 344, 321, 327, 381, 327

The trajectory of 9 is 23 steps in length and overlaps the trajectory of 1 (shown in blue):

9, 90, 171, 222, 210, 207, 252, 269, 310, 320, 325, 355, 414, 383, 337, 404, 372, 426, 370, 428, 344, 321, 327, 381, 327

14 is the next number of interest:

Without listing the trajectories for 10, 11, 12 and 13, it can be noted that all of their trajectories overlap that of 1. However, once we reach 14, there is a new development. The trajectory is 14, -1, 0. What happens of course is that -1 is reached and after that the trajectory is stuck on 0. See Figure 2.

Figure 2

22 and 42  are the next numbers of interest:

After this the trajectories for 15, 16, 17, 18, 19, 20 and 21 all overlap that of 1. Once 22 is reached, the same situation as with 14 prevails. The trajectory of 22 is 22, 14, -1, 0. After 22, all trajectories overlap again with that of 1 until 42 is reached and the trajectory once again plummets to zero: 42, 22, 14, -1, 0, 0.

50 is the next number of interest:

It is only when 50 is reached that we get a new trajectory. See Figure 3.

Figure 3

Once again a loop is reached but this time it is {573, 656, 609, 654, 627, 636}. The full trajectory is:

50, 75, 149, 215, 237, 291, 369, 423, 412, 393, 492, 553, 612, 573, 656, 609, 654, 627, 636, 573

62 and 75 are the next numbers of interest:

From 51 to 61, the trajectories again overlap that of 1 but at 62 it plummets to zero with a trajectory of 62, 22, 14, -1, 0, 0. From 62 to 74, the trajectory overlaps that of 1 until, at 75, the trajectory overlaps that of 50 as can be expected because 50 --> 75:

75, 149, 215, 237, 291, 369, 423, 412, 393, 492, 553, 612, 573, 656, 609, 654, 627, 636, 573

82 and 92 are the next numbers of interest:

From 76 to 81 we're back to overlapping the trajectory of 1 and at 82 we go to zero again with 82, 14, -1, 0, 0. From 83 to 91, we are back to overlapping the trajectory of 1 but at 92 we overlap the trajectory of 50 and 75 (shown in blue):

92, 169, 215, 237, 291, 369, 423, 412, 393, 492, 553, 612, 573, 656, 609, 654, 627, 636, 573

The remaining trajectories for 93 to 99 all overlap the trajectory of 1. These results for the numbers 1 to 99 can be summarised as follows:

• 50, 75 and 92 end in the loop {573, 656, 609, 654, 627, 636}
• 14, 22, 62 and 82 end in 0
• all other numbers end in the loop {327, 381}

The general observation can be made that with \(k=2\) the numbers in the trajectory sequence will eventually rise because any negative odd digits (odd digits always predominate), will become positive when squared. This is not the case of course when \(k=3\) and we will look at this in a subsequent post.

Higher Order Odds and Evens Trajectory

My previous post on higher order Smith numbers gave me the idea to look into what happens with the odd and even trajectory of numbers when the squares of the digits are added or subtracted instead of just the digits themselves. The odd and even rule applied repeatedly to a number \(n\) is simply to add to \(n\) its odd digits and subtract its even ones. I've devoted numerous posts to the analysis of what happens beginning with my post titled Odds and Evens on June 17th 2021.

Numbers whose sum of odd and even digits is equal are not altered when this rule is applied. 112 is an example of such a number and numbers of this type I termed attractors. Other numbers follow a trajectory that either leads them to an attractor or else they enter a loop. Such a loop I termed a vortex and the numbers that comprise a vortex I termed vorticals. 

101 is an example of a number whose trajectory leads to an attractor, in this case 134. Here is its trajectory: 101, 103, 107, 115, 122, 119, 130, 134. 199 is an example of a number that enters a vortex. Its trajectory is 199, 218, 209, 216, 209. Here we see that {209, 216} is the vortex and the vorticals are 209 and 216. Numbers whose trajectories lead to attractors or vortices I've termed captives.

The purpose of this post is to explore what happens when, instead of adding just the odd digit or subtracting the even digit, we add higher powers of those digits. Let's begin by looking at what happens to the number associated with my diurnal age today viz. 26638. Under the odd and even rule, the trajectory is 26638, 26619, 26615, 26607, 26600, 26586, 26569. The trajectory leads to the attractor 26569 that has a total of 92 captives.

What if the squares of the digits are added and subtracted? Well the trajectory then becomes 26638, 26507, 26541, 26511, 26498, 26459, 26509, 26575, 26634, 26551, 26562, 26507 and it can be seen that the trajectory immediately enters a loop or vortex {26507, 26541, 26511, 26498, 26459, 26509, 26575, 26634, 26551, 26562}. 

Let's try with the cubes of the digits. In this case, we end up with an impressively long trajectory:

26638, 25713, 26201, 25970, 27159, 28349, 28521, 28119, 28330, 27864, 27407, 28021, 27494, 28430, 27873, 28066, 27114, 27387, 27580, 27528, 27468, 27011, 27348, 27134, 27433, 27758, 28049, 28194, 28340, 27783, 27976, 29167, 30016, 29828, 29517, 30707, 31420, 31376, 31558, 31324, 31307, 31705, 32201, 32213, 32252, 32380, 31914, 32608, 31899, 32873, 32750, 33237, 33653, 33643, 33444, 33306, 33171, 33570, 34092, 34776, 35209, 36082, 35373, 35922, 36787, 36772, 37261, 37408, 37202, 37556, 37960, 38843, 37809, 38396, 38451, 38028, 37023, 37412, 37711, 38426, 37653, 37959, 39912, 41390, 42083, 41526, 41364, 41048, 40409, 41010, 40948, 41037, 41344, 41180, 40606, 40110, 40048, 39408, 39588, 39445, 40198, 40352, 40432, 40323, 40305, 40393, 41112, 41043, 40943, 41571, 41977, 43329, 44040, 43848, 42723, 43013, 43004, 42903, 43587, 43506, 43378, 43199, 44621, 44270, 44477, 44971, 45916, 46491, 46877, 46771, 47178, 47289, 47777, 49085, 49363, 49866, 49587, 50208, 49813, 49994, 52053, 52322, 52450, 52628, 52009, 52855, 52710, 53171, 53668, 52876, 52608, 51997, 53924, 54733, 55191, 56172, 56417, 56606, 56083, 55507, 56225, 56243, 56107, 56360, 56080, 55477, 56349, 56950, 57713, 58552, 58407, 58299, 59362, 60019, 60533, 60496, 60729, 61577, 62173, 62320, 62115, 62018, 61283, 60575, 60952, 61582, 60972, 61820, 61085, 60483, 59718, 60404, 60060, 59628, 59746, 60663, 60042, 59754, 61012, 60790, 61646, 60935, 61600, 61169, 61468, 60461, 59966, 61117, 61247, 61303, 61142, 60856, 60037, 60191, 60706, 60617, 60529, 61159, 61799, 63385, 62836, 61911, 62427, 62474, 62465, 62086, 61134, 60883, 59670, 60651, 60345, 60217, 60337, 60518, 59916, 61284, 60485, 59818, 59649, 60952

As can be seen, the trajectory of 238 steps finally enters a loop or vortex {60952 ... 59649}. Figure 1 shows a plot of the trajectory.

Figure 1: permalink

When we try with fourth powers we find that the trajectory consists of 599 steps:

26638, 20015, 20625, 19922, 33013, 33257, 36429, 41503, 41954, 48629, 49526, 55144, 55883, 49022, 55295, 63715, 65527, 67866, 62283, 56940, 62574, 64032, 62545, 62227, 63284, 57701, 63129, 68460, 61516, 59551, 67988, 67462, 66999, 84090, 86299, 94013, 100400, 100145, 100516, 99847, 111018, 106926, 110880, 102690, 107940, 116647, 116202, 114876, 111631, 110420, 110150, 110778, 111486, 105841, 102116, 100807, 99113, 112318, 108290, 110740, 112887, 107082, 105372, 108464, 102561, 101876, 98887, 95561, 102077, 106864, 99921, 119589, 129242, 135500, 136832, 131587, 130600, 129386, 130621, 129392, 142564, 141366, 138601, 133292, 139984, 148836, 139174, 147963, 155455, 157700, 163128, 157803, 156815, 152675, 155015, 156892, 158671, 156307, 158119, 161212, 159887, 161283, 155958, 160299, 172110, 174498, 178853, 173769, 183918, 182370, 180741, 178792, 186044, 180141, 175792, 187765, 187801, 182012, 177886, 173201, 175669, 182665, 176587, 176623, 176498, 179813, 184762, 181500, 178031, 176419, 183831, 175803, 174815, 173491, 182280, 174057, 179229, 194721, 203413, 203304, 203194, 209565, 216064, 213201, 213252, 213911, 220540, 220877, 221551, 222771, 227526, 229208, 231625, 231004, 230814, 226528, 221713, 224165, 223207, 225641, 224683, 219084, 221278, 219536, 225492, 232374, 234649, 239467, 246942, 251663, 249762, 257140, 259895, 270155, 273791, 285220, 281701, 279992, 302044, 301597, 311266, 308741, 306872, 303946, 309117, 318162, 312837, 311289, 313821, 309873, 314901, 321289, 323804, 319598, 329331, 336120, 334971, 343840, 339394, 352503, 353899, 363712, 364964, 368502, 363800, 358570, 358206, 353504, 354660, 352518, 349738, 354510, 355586, 352150, 353466, 351405, 352481, 348820, 340437, 342488, 333849, 336301, 335249, 342325, 342824, 338265, 333644, 332079, 341187, 339319, 352685, 348608, 338945, 341941, 348073, 346284, 340445, 340383, 336274, 337269, 345081, 341436, 339791, 355477, 361354, 360590, 366561, 363380, 358231, 354907, 364319, 369491, 381143, 376955, 385952, 389732, 394744, 403019, 409406, 414159, 420835, 417173, 421802, 417419, 425871, 424530, 424708, 422485, 418470, 416264, 413145, 413341, 412993, 425925, 433448, 428746, 425227, 427949, 442944, 448465, 442930, 449044, 454581, 451224, 451306, 450461, 449279, 464274, 464595, 470598, 475833, 474669, 480527, 479185, 484421, 479542, 488601, 478858, 469340, 474174, 478209, 482803, 474420, 476037, 479368, 482763, 479581, 484817, 478515, 477815, 478891, 479406, 486560, 480241, 475618, 472997, 490649, 501963, 507935, 518228, 510630, 510041, 510412, 510767, 514899, 524295, 531818, 524334, 524593, 532213, 532969, 545485, 542752, 546115, 545815, 543339, 550512, 552372, 556072, 558411, 555311, 557269, 566169, 569468, 569710, 578002, 576916, 583912, 587068, 580606, 574543, 577763, 584376, 581835, 574975, 587332, 586408, 577289, 585165, 581649, 583188, 571607, 575739, 588433, 580772, 582087, 576905, 585821, 578864, 572146, 573605, 576041, 577516, 582273, 581252, 578375, 580412, 576670, 579505, 590342, 597337, 609487, 612801, 607395, 615767, 618603, 611997, 626226, 622290, 627507, 631622, 629080, 630233, 629164, 632862, 626223, 623664, 619585, 622005, 621302, 620056, 618073, 615164, 612943, 618018, 608532, 603830, 598600, 600394, 605484, 600205, 599518, 609795, 624647, 623928, 625146, 622908, 624045, 622846, 615870, 613505, 613541, 612697, 619052, 624927, 632305, 631780, 628871, 621769, 628124, 622445, 621230, 619984, 627459, 635478, 632937, 640749, 647903, 655394, 661734, 661369, 664124, 661005, 659039, 671571, 675704, 679579, 696832, 696770, 705541, 708937, 716285, 713904, 722692, 730310, 732874, 733389, 738498, 739093, 754778, 758254, 757537, 766071, 768282, 761163, 761055, 763411, 764343, 765098, 769293, 783585, 779125, 791098, 802526, 797727, 813876, 806871, 799785, 814238, 805856, 797618, 803590, 806761, 802475, 801133, 797201, 808549, 807287, 803881, 791675, 802368, 792945, 808821, 796518, 800714, 798764, 804479, 808833, 796707, 809175, 814667, 810125, 806640, 799696, 819188, 813463, 807978, 811149, 813361, 808133, 800104, 795753, 808447, 802144, 797521, 809494, 818008, 805721, 804636, 797773, 814019, 816230, 810904, 813114, 808846, 795006, 803297, 808228, 795908, 807960, 811530, 808142, 799679, 822868, 809252, 812310, 808281, 795978, 810431, 806162, 799459, 821912, 824347, 822205, 818686, 803807, 798097, 811925, 815001, 811532, 808128, 795825, 801925, 805000, 801529, 804604, 798700, 805967, 810162, 804756, 802134, 797848, 800763, 797853, 805826, 796947, 813319, 815948, 814687, 807345, 806100, 800709, 805575, 805755, 805935, 809731, 814679, 817994, 829166, 829024, 831201, 827172, 827847, 824185, 816347, 813182, 805057, 804612, 798949, 816681, 805899, 811454, 807473, 808004, 799556, 815033, 811725, 810641, 804995, 814390, 816681

Once again, a loop or vortex is entered {816681 ... 814390}. Figure 2 shows a plot of the trajectory.

Figure 2: permalink

With fifth powers, the trajectory is even longer (971 steps) and, as can be seen, the trajectory initially plunges into negative number territory.

26638, -21471, -37224, -53186, -16011, -8238, 57087, 61058, 23640, 15051, 21303, 21758, 8891, 2405, 4474, 18209, 44459, 103561, 99155, 223504, 225784, 211860, 171286, 147519, 225478, 211554, 216750, 228875, 183207, 167458, 145823, 115368, 78194, 120259, 182370, 166621, 143263, 134918, 160420, 151589, 184122, 150268, 112818, 47253, 66372, 67838, 11576, 23734, 39971, 175120, 195022, 257133, 277520, 314195, 375590, 457939, 595188, 594952, 718244, 700204, 715955, 801187, 752460, 763560, 768183, 711922, 787716, 797594, 951407, 1029365, 1083975, 1130432, 1129864, 1147315, 1166469, 1201168, 1160595, 1218120, 1185291, 1214668, 1165294, 1218638, 1145539, 1210059, 1272203, 1289158, 1285766, 1257347, 1293274, 1368286, 1287410, 1270395, 1349588, 1345446, 1337967, 1423341, 1421749, 1495527, 1576578, 1575899, 1684287, 1626727, 1644726, 1643902, 1694363, 1737323, 1771635, 1800843, 1734527, 1770454, 1805146, 1766705, 1787893, 1815264, 1776791, 1878487, 1812774, 1812566, 1767341, 1792400, 1867201, 1843434, 1808081, 1709779, 1878299, 1947637, 2031744, 2046715, 2057816, 2037173, 2071242, 2086930, 2105646, 2092164, 2142350, 2144631, 2135020, 2138325, 2109105, 2171249, 2246019, 2296205, 2350507, 2373775, 2427775, 2480233, 2446863, 2396706, 2457221, 2476034, 2483228, 2416815, 2378342, 2361779, 2446878, 2388293, 2382228, 2316807, 2293282, 2319678, 2355202, 2361599, 2475258, 2464459, 2515753, 2542147, 2559968, 2643740, 2650934, 2704519, 2782445, 2767497, 2868135, 2798160, 2833441, 2799080, 2901185, 2930561, 2985171, 3031354, 3034185, 3004005, 3006349, 3057084, 3043467, 3050936, 3105820, 3076389, 3112187, 3096440, 3145908, 3174534, 3192905, 3314340, 3313022, 3313688, 3241106, 3232519, 3295116, 3349727, 3441820, 3407216, 3415435, 3420124, 3418256, 3380025, 3350836, 3314146, 3304810, 3271505, 3294774, 3385600, 3348667, 3316616, 3293776, 3379117, 3472268, 3447686, 3414368, 3372263, 3381959, 3470901, 3545977, 3644109, 3693578, 3732501, 3752888, 3674727, 3716559, 3791133, 3867720, 3861001, 3820702, 3804920, 3830388, 3732813, 3717550, 3757658, 3757221, 3794140, 3868192, 3854141, 3822695, 3844504, 3812032, 3779687, 3848856, 3745120, 3764240, 3771434, 3803487, 3754220, 3773307, 3824457, 3809784, 3819323, 3846302, 3805188, 3710253, 3730640, 3739133, 3815962, 3837804, 3788537, 3760226, 3761660, 3755383, 3746401, 3753628, 3733470, 3766789, 3811375, 3799027, 3950950, 4075541, 4096551, 4153051, 4158522, 4130917, 4205994, 4325137, 4344500, 4344796, 4410047, 4423783, 4406228, 4363572, 4375158, 4364667, 4356341, 4350129, 4411491, 4467471, 4490238, 4514682, 4475184, 4459277, 4552985, 4587585, 4547207, 4581866, 4502880, 4439413, 4495877, 4556849, 4579556, 4655987, 4696525, 4745216, 4755293, 4836586, 4757842, 4759733, 4854983, 4849816, 4833506, 4795549, 4934656, 4979473, 5129380, 5158998, 5217811, 5204946, 5257264, 5271457, 5310266, 5298051, 5330551, 5340413, 5341977, 5436985, 5460959, 5576507, 5611720, 5623846, 5577838, 5552409, 5619777, 5724597, 5822454, 5793824, 5839224, 5867785, 5834337, 5821206, 5783724, 5786882, 5700702, 5737409, 5832416, 5794185, 5842500, 5814926, 5835501, 5812352, 5786014, 5764379, 5851610, 5817318, 5771959, 5929922, 6110098, 6128605, 6083379, 6119177, 6204067, 6204266, 6179850, 6218288, 6112145, 6106441, 6088843, 5981982, 6037638, 6006611, 5983285, 5983259, 6075050, 6090331, 6142091, 6192310, 6243796, 6303287, 6280004, 6238404, 6196023, 6239732, 6308234, 6267120, 6268312, 6220172, 6229108, 6247550, 6261775, 6282931, 6301616, 6278533, 6258375, 6241099, 6350366, 6330649, 6373608, 6342581, 6304350, 6299161, 6401677, 6418716, 6386181, 6305338, 6268648, 6178728, 6138999, 6275846, 6246402, 6228738, 6172412, 6180357, 6159989, 6299718, 6394048, 6410748, 6384964, 6393888, 6347343, 6355055, 6360022, 6344649, 6385317, 6365192, 6412026, 6395387, 6434310, 6424973, 6491216, 6533659, 6583892, 6572965, 6639487, 6666242, 6634050, 6620842, 6571434, 6581786, 6520631, 6508416, 6462198, 6471872, 6463887, 6398825, 6387898, 6357917, 6446173, 6445624, 6430093, 6480828, 6373692, 6434450, 6426970, 6486218, 6404075, 6414183, 6371836, 6340810, 6299486, 6368208, 6287331, 6264049, 6305466, 6284482, 6209058, 6230656, 6210664, 6186281, 6105163, 6092981, 6170504, 6181637, 6150369, 6197235, 6268652, 6215617, 6219967, 6339289, 6417297, 6501129, 6555497, 6631928, 6642869, 6644766, 6628421, 6579014, 6649196, 6742943, 6809186, 6787148, 6746427, 6762409, 6821657, 6793238, 6829004, 6846453, 6799453, 6928926, 6998640, 7067394, 7151500, 7174559, 7272449, 7363000, 7372517, 7426275, 7454150, 7475160, 7503100, 7523276, 7552418, 7541652, 7555878, 7533331, 7554236, 7568704, 7563875, 7563438, 7542288, 7495596, 7627951, 7715932, 7811932, 7855233, 7845976, 7900196, 8027326, 8003768, 7947506, 8034494, 8057946, 8095359, 8187182, 8105655, 8074487, 8040517, 8026658, 7948663, 7975418, 8037415, 8023799, 8126147, 8101356, 8064182, 7989815, 8062310, 8021978, 8032267, 8008709, 8019029, 8104328, 8037980, 8048543, 7984327, 8043409, 8067885, 7981737, 8058683, 7955971, 8113934, 8139679, 8234284, 8166879, 8161648, 8079538, 8093226, 8111910, 8138195, 8135078, 8089718, 8067271, 8060310, 8020010, 7987211, 8047076, 8039122, 8065583, 7998764, 8108908, 8069654, 8082484, 7982100, 8025157, 8015415, 7987875, 8034934, 8059653, 8084651, 8013441, 7978870, 8022804, 7956180, 7994618, 8087956, 8093625, 8115466, 8069249, 8145747, 8147671, 8139719, 8242101, 8208247, 8158430, 8095239, 8183905, 8180787, 8116098, 8101837, 8053353, 8027564, 8005896, 7994758, 8115803, 8053637, 8033511, 8004356, 7966156, 8021810, 7956244, 8025369, 8047210, 8030194, 8055695, 8083575, 8041339, 8067083, 8010821, 7945255, 8029430, 8054898, 8017744, 8016543, 7978344, 8036434, 7994328, 8095652, 8120375, 8107751, 8111724, 8094710, 8136775, 8133214, 8099878, 8136479, 8171011, 8155054, 8130638, 8057813, 8012453, 7981998, 8110417, 8093435, 8122303, 8089958, 8112877, 8080925, 8077531, 8081746, 8024218, 7957595, 8118682, 8012572, 7999673, 8202901, 8229119, 8314387, 8265121, 8227640, 8202815, 8140341, 8105770, 8109742, 8151775, 8158873, 8080745, 8034117, 8017377, 8035274, 8021625, 7984143, 8025427, 8011503, 7982105, 8028287, 7946726, 8022781, 7973989, 8152225, 8125612, 8088131, 7990072, 8141752, 8127862, 8071294, 8113327, 8097822, 8108078, 8026582, 7956331, 8028023, 7962666, 8007386, 7951124, 8029051, 8058426, 7987183, 8014554, 7985989, 8117532, 8104909, 8189216, 8174923, 8217199, 8319306, 8338298, 8299497, 8459627, 8497008, 8506304, 8468104, 8392745, 8438145, 8373930, 8417747, 8433353, 8403658, 8332690, 8351649, 8372499, 8473823, 8424524, 8391745, 8437178, 8404476, 8377667, 8380011, 8314720, 8297947, 8415835, 8355769, 8397574, 8459813, 8455671, 8437161, 8412645, 8373147, 8373456, 8352306, 8315341, 8285162, 8214912, 8240107, 8223091, 8249552, 8280995, 8336650, 8291941, 8376217, 8369499, 8505321, 8479015, 8524205, 8496599, 8635303, 8598613, 8587719, 8617972, 8670060, 8638547, 8584386, 8480650, 8409439, 8492964, 8568438, 8464702, 8438885, 8310157, 8297566, 8328195, 8325045, 8297714, 8356554, 8324604, 8282223, 8216802, 8143427, 8125630, 8088423, 7989306, 8083910, 8077667, 8079768, 8099119, 8243500, 8213044, 8178440, 8127664, 8095096, 8175775, 8199679, 8353090, 8382982, 8343906, 8361873, 8305855, 8249937, 8351261, 8314055, 8286757, 8250152, 8223571, 8210915, 8240291, 8265485, 8197367, 8249730, 8292005, 8321347, 8304817, 8255308, 8196233, 8215193, 8244812, 8177165, 8173362, 8150080, 8087670, 8047972, 8106811, 8033502, 8004313, 7971008, 8030904, 8056404, 8016937, 8052493, 8081086, 7975007, 8087602, 8031065, 7993890, 8155319, 8188095, 8151966, 8165822, 8095572, 8144878, 8061334, 8020253, 7990789, 8168782, 8079478, 8105581, 8046297, 8080553, 8021510, 7991837, 8111025, 8081353, 8019429, 8103704, 8086963, 8065167, 8036780, 7980518, 7993964, 8179361, 8214918, 8207377, 8225241, 8194479, 8294569, 8374192, 8416468, 8333333, 8302023, 8269677, 8313988, 8275220, 8262288, 8156112, 8118664, 8036554, 8001479, 8043544, 8011072, 7995081, 8100344, 8065772, 8061935, 8083809, 8044797, 8102644, 8060021, 8019446, 8035904, 8064529, 8085103, 8022936, 8041620, 8000021, 7967222, 8052013, 8022582, 7960075, 8048087, 7965566, 8024344, 7988715, 8018968, 7971938, 8091126, 8109601, 8128108, 8029774, 8088613, 7982777, 8076254, 8054586, 7986500, 8024937, 8067212, 8043412, 8008808, 7877736, 7904663, 7964186, 7990699, 8235926, 8257735, 8265042, 8226535, 8192420, 8217614, 8192823, 8186516, 8108555, 8052395, 8085137, 8039777, 8116722, 8092923, 8178432, 8128891, 8089606, 8067567, 8055986, 8047973, 8107087, 8075166, 8046779, 8097874, 8123977, 8184084, 8083733, 8035733, 8023626, 7975485, 8040606, 7991262, 8118328, 8020237, 8004455, 7975889, 8065190, 8086821, 7980710, 8040606

The trajectory once again enters a loop {8040606 ... 7980710}. Figure 3 shows the trajectory.

Figure 3: permalink

I'll leave off there and powers higher than 5 may prove rather unwieldy. It's not possible to make generalisations at this early stage but it would seem that the trajectories are far more volatile and lengthy once we start dealing with powers of digits. The trajectories may end up favouring vortices instead of attractors. 

One general observation is that the trajectories will always tend to rise rather than fall because there are the odd digits (1, 3, 5, 7 and 9) with an average value of 5 outweighing the even digits (0, 2, 4, 6, 8) with an average value of 4.