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.