The original idea for Kaprekar numbers defined them roughly as follows:

Definition A: N is a Kaprekar number if it has D digits, and if you take N² and divide it into two pieces each D digits in size and add them together, you get N.

For example: 45² = 2025, and 20+25 = 45, so 45 is a Kaprekar number.

Here is an example of a Kaprekar number for cubes: 7172³ = 368910352448 and 3689+1035+2448 = 7172. There are also solutions for higher powers.

On this page Douglas E. Iannucci explains how the Kaprekar numbers (for squares) can be derived from the prime factorization of 10^D-1. He also shows that if N is a Kaprekar number of D digits, then 10^D-N is also a Kaprekar number of D digits. Thus, for example, since 45 is a 2-digit Kaprekar number we know that 55 is as well because 45+55 = 100. However, this only holds true if you allow Kaprekar numbers to start with zero, such as the "five-digit" example 04879²=0023804641, 00238+04641=04879 (with small 0's added to show that 4879 is being treated as a 5-digit number). If such numbers were not allowed, the symmetry would break down: 95121 is also a Kaprekar number and 95121+04879 = 10⁵.

Because of this, the definition of the Kaprekar numbers is usually given somewhat like this:

Definiton B: N is a Kaprekar number if there are three numbers D>0, 0<=Q and 0<=R<10^D
such that N²=Q×10^D+R and Q+R=N.

This definition allows the 04879² solution, and also allows the solution N=10, D=1, Q=10, R=0 (which takes 10²=100 and divides it into two pieces like this: 10 0).

Although the reasons for Definition B are well-justified by the mathematics showing how to derive all the Kaprekar numbers, these reasons disappear when we discuss Kaprekar numbers for cubes and higher powers, where the aforementioned symmetry does not exist. Also, Definition B does not follow the intuitive nature of definition A: D should be the number of digits in N, neither more (as with 04879) nor less (as with the bogus 10²=100 example). The definition can be brought in line with Definition A by explicitly stating that D is the number of digits in N:

Definiton C: N is a Kaprekar number if there are three numbers D>0, 0<=Q and 0<=R<10^D
such that 10^D-1 <= N < 10^D, N²=Q×10^D+R and Q+R=N.

In order to prevent further confusion I will call these the "natural Kaprekar numbers".

There are also definitions that add additional requirements for R, and perhaps Q as well, to be D-digit numbers. They are apparently intended to prevent the 10²=10 0 case, but also prevent legitimate Kaprekar numbers like 999²=998001, 998+001=999 and 297³ = 26198073, 26+198+073 = 297.

Not all of the variations of each Kaprekar sequence are included in Sloane's catalog. In particular, the natural versions are lacking or labeled "incorrect". Since the other sequences can very easily be derived from the natural sequence, I will give only that version.

Here are the natural Kaprekar numbers:

For squares: 1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4950, 5050, 7272, 7777, 9999, 17344, 22222, 77778, 82656, 95121, 99999, 142857, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170, 538461, 609687, 643357, 648648, 670033, 681318, 791505, 812890, 818181, 851851, 857143, 961038, 994708, 999999, 4444444, 4927941, 5072059, 5555556, 9372385, 9999999, 11111112, 13641364, 16590564, 19273023, 19773073, 24752475, 25252525, 30884184, 36363636, 38883889, 44363341, 44525548, 49995000, 50005000, 55474452, 55636659, 61116111, 63636364, 69115816, 74747475, 75247525, 80226927, 80726977, 83409436, 86358636, 88888888, 91838088, 94520547, 99999999, 234567901, 332999667, 432432432, 567567568, 667000333, 765432099, 999999999, 1111111111, 1776299581, 2020202020, 3846956652, 3888938889, 4090859091, 4132841328, ...

For 3^rd powers: 1, 8, 10, 45, 297, 2322, 2728, 4445, 4544, 4949, 5049, 5455, 5554, 7172, 27100, 44443, 55556, 60434, 77778, 143857, 208494, 226071, 279720, 313390, 324675, 329967, 346060, 368928, 395604, 422577, 427868, 461539, 472823, 478115, 488214, 494208, 495208, 499500, 500500, 517076, 533170, 543752, 559846, 565137, 598807, 664741, 670032, 720279, 757835, 791505, 807598, 825175, 829466, 856142, 966329, 973323, 4444443, 4927940, 5072058, 5555555, 5555556, 5699673, 6183170, 8888887, 11273318, 13793570, 17090613, 21803275, 22293325, 24752475, 25242525, 25252524, 27272728, 27282727, 28201724, 30731977, 33404436, 36363635, 38383839, 38546045, 38883889, 39046095, 39546145, 41843088, 44025497, 44363340, 44525547, 45025597, 47045800, 49842793, 49995000, 50005000, 50657256, 55474451, 55474452, 55484452, 56136708, 58156911, 58656961, 60453854, 61453954, 63636364, 63798570, 64298620, 66747770, 69278022, 69768072, 69778072, 72227222, 72717272, 72727271, 74747475, 74909681, 75409731, 77706674, 77767777, 77777776, 80726976, 80726977, 80889183, 88878888, 88888888, 93868290, 197864531, 332999667, 667000332, ...

For 4^th powers: 1, 7, 45, 55, 67, 100, 433, 4950, 5050, 38212, 65068, 190576, 295075, 299035, 310024, 336700, 343333, 394615, 414558, 433566, 448228, 450550, 467236, 475497, 476191, 486486, 499500, 500500, 523513, 534898, 549550, 599743, 622414, 628408, 647362, 652015, 671671, 677755, 705331, 731368, 734932, 765963, 772200, 803539, 820612, 826606, 849520, 934066, 25280200, 33333334, 44525548, 49995000, 50005000, 55474452, 58585858, 61643836, 77696674, 332999667, 334333333, 811144477, 3577235773, ...

For 5^th powers: 1, 10, 1000, 7776, 27100, 73440, 95120, 500499, 505791, 540539, 598697, 665335, 697598, 732347, 7607610, 37944478, 46945205, 54995500, 55216205, 56607166, ...

Table of expansions of normal (order-2) Kaprekar numbers:

1² = 1 ; 0 + 1 = 0
9² = 81 ; 8 + 1 = 8
45² = 2025 ; 20 + 25 = 20
55² = 3025 ; 30 + 25 = 30
99² = 9801 ; 98 + 01 = 98
297² = 88209 ; 088 + 209 = 88
703² = 494209 ; 494 + 209 = 494
999² = 998001 ; 998 + 001 = 998
2223² = 4941729 ; 0494 + 1729 = 494
2728² = 7441984 ; 0744 + 1984 = 744
4950² = 24502500 ; 2450 + 2500 = 2450
5050² = 25502500 ; 2550 + 2500 = 2550
7272² = 52881984 ; 5288 + 1984 = 5288
7777² = 60481729 ; 6048 + 1729 = 6048
9999² = 99980001 ; 9998 + 0001 = 9998
17344² = 300814336 ; 03008 + 14336 = 3008
22222² = 493817284 ; 04938 + 17284 = 4938
77778² = 6049417284 ; 60494 + 17284 = 60494
82656² = 6832014336 ; 68320 + 14336 = 68320
95121² = 9048004641 ; 90480 + 04641 = 90480
99999² = 9999800001 ; 99998 + 00001 = 99998
142857² = 20408122449 ; 020408 + 122449 = 20408
148149² = 21948126201 ; 021948 + 126201 = 21948
181819² = 33058148761 ; 033058 + 148761 = 33058
187110² = 35010152100 ; 035010 + 152100 = 35010
208495² = 43470165025 ; 043470 + 165025 = 43470
318682² = 101558217124 ; 101558 + 217124 = 101558
329967² = 108878221089 ; 108878 + 221089 = 108878
351352² = 123448227904 ; 123448 + 227904 = 123448
356643² = 127194229449 ; 127194 + 229449 = 127194
390313² = 152344237969 ; 152344 + 237969 = 152344
461539² = 213018248521 ; 213018 + 248521 = 213018
466830² = 217930248900 ; 217930 + 248900 = 217930
499500² = 249500250000 ; 249500 + 250000 = 249500
500500² = 250500250000 ; 250500 + 250000 = 250500
533170² = 284270248900 ; 284270 + 248900 = 284270
538461² = 289940248521 ; 289940 + 248521 = 289940
609687² = 371718237969 ; 371718 + 237969 = 371718
643357² = 413908229449 ; 413908 + 229449 = 413908
648648² = 420744227904 ; 420744 + 227904 = 420744
670033² = 448944221089 ; 448944 + 221089 = 448944
681318² = 464194217124 ; 464194 + 217124 = 464194
791505² = 626480165025 ; 626480 + 165025 = 626480
812890² = 660790152100 ; 660790 + 152100 = 660790
818181² = 669420148761 ; 669420 + 148761 = 669420
851851² = 725650126201 ; 725650 + 126201 = 725650
857143² = 734694122449 ; 734694 + 122449 = 734694
961038² = 923594037444 ; 923594 + 037444 = 923594
994708² = 989444005264 ; 989444 + 005264 = 989444
999999² = 999998000001 ; 999998 + 000001 = 999998
4444444² = 19753082469136 ; 1975308 + 2469136 = 1975308
4927941² = 24284602499481 ; 2428460 + 2499481 = 2428460
5072059² = 25725782499481 ; 2572578 + 2499481 = 2572578
5555556² = 30864202469136 ; 3086420 + 2469136 = 3086420
9372385² = 87841600588225 ; 8784160 + 0588225 = 8784160
9999999² = 99999980000001 ; 9999998 + 0000001 = 9999998
11111112² = 123456809876544 ; 01234568 + 09876544 = 1234568
13641364² = 186086811780496 ; 01860868 + 11780496 = 1860868
16590564² = 275246813838096 ; 02752468 + 13838096 = 2752468
19273023² = 371449415558529 ; 03714494 + 15558529 = 3714494
19773073² = 390974415863329 ; 03909744 + 15863329 = 3909744
24752475² = 612685018625625 ; 06126850 + 18625625 = 6126850
25252525² = 637690018875625 ; 06376900 + 18875625 = 6376900
30884184² = 953832821345856 ; 09538328 + 21345856 = 9538328
36363636² = 1322314023140496 ; 13223140 + 23140496 = 13223140
38883889² = 1511956823764321 ; 15119568 + 23764321 = 15119568
44363341² = 1968106024682281 ; 19681060 + 24682281 = 19681060
44525548² = 1982524424700304 ; 19825244 + 24700304 = 19825244
49995000² = 2499500025000000 ; 24995000 + 25000000 = 24995000
50005000² = 2500500025000000 ; 25005000 + 25000000 = 25005000
55474452² = 3077414824700304 ; 30774148 + 24700304 = 30774148
55636659² = 3095437824682281 ; 30954378 + 24682281 = 30954378
61116111² = 3735179023764321 ; 37351790 + 23764321 = 37351790
63636364² = 4049586823140496 ; 40495868 + 23140496 = 40495868
69115816² = 4776996021345856 ; 47769960 + 21345856 = 47769960
74747475² = 5587185018875625 ; 55871850 + 18875625 = 55871850
75247525² = 5662190018625625 ; 56621900 + 18625625 = 56621900
80226927² = 6436359815863329 ; 64363598 + 15863329 = 64363598
80726977² = 6516844815558529 ; 65168448 + 15558529 = 65168448
83409436² = 6957134013838096 ; 69571340 + 13838096 = 69571340
86358636² = 7457814011780496 ; 74578140 + 11780496 = 74578140
88888888² = 7901234409876544 ; 79012344 + 09876544 = 79012344
91838088² = 8434234407495744 ; 84342344 + 07495744 = 84342344
94520547² = 8934133805179209 ; 89341338 + 05179209 = 89341338
99999999² = 9999999800000001 ; 99999998 + 00000001 = 99999998
234567901² = 55022100179545801 ; 055022100 + 179545801 = 55022100
332999667² = 110888778222110889 ; 110888778 + 222110889 = 110888778
432432432² = 186997808245434624 ; 186997808 + 245434624 = 186997808
567567568² = 322132944245434624 ; 322132944 + 245434624 = 322132944
667000333² = 444889444222110889 ; 444889444 + 222110889 = 444889444
765432099² = 585886298179545801 ; 585886298 + 179545801 = 585886298
999999999² = 999999998000000001 ; 999999998 + 000000001 = 999999998
1111111111² = 1234567900987654321 ; 0123456790 + 0987654321 = 123456790
1776299581² = 3155240201460775561 ; 0315524020 + 1460775561 = 315524020
2020202020² = 4081216201612080400 ; 0408121620 + 1612080400 = 408121620
3846956652² = 14799075482367049104 ; 1479907548 + 2367049104 = 1479907548
3888938889² = 15123845682376554321 ; 1512384568 + 2376554321 = 1512384568
4090859091² = 16735128102417346281 ; 1673512810 + 2417346281 = 1673512810
4132841328² = 17080377442424803584 ; 1708037744 + 2424803584 = 1708037744
(checked all bases through 4294967294)

Table of expansions of order-3 Kaprekar numbers:

1³ = 1 ; 0+0+1 = 1
8³ = 512 ; 5+1+2 = 8
10³ = 1000 ; 0+10+00 = 10
45³ = 91125 ; 9+11+25 = 45
297³ = 26198073 ; 26+198+073 = 297
2322³ = 12519490248 ; 125+1949+0248 = 2322
2728³ = 20301732352 ; 203+0173+2352 = 2728
4445³ = 87824421125 ; 878+2442+1125 = 4445
4544³ = 93824221184 ; 938+2422+1184 = 4544
4949³ = 121213882349 ; 1212+1388+2349 = 4949
5049³ = 128711132649 ; 1287+1113+2649 = 5049
5455³ = 162324571375 ; 1623+2457+1375 = 5455
5554³ = 171323771464 ; 1713+2377+1464 = 5554
7172³ = 368910352448 ; 3689+1035+2448 = 7172
27100³ = 19902511000000 ; 1990+25110+00000 = 27100
44443³ = 87782935806307 ; 8778+29358+06307 = 44443
55556³ = 171471879319616 ; 17147+18793+19616 = 55556
60434³ = 220721185826504 ; 22072+11858+26504 = 60434
77778³ = 470511577514952 ; 47051+15775+14952 = 77778
143857³ = 2977097087043793 ; 2977+097087+043793 = 143857
208494³ = 9063181647017784 ; 9063+181647+017784 = 208494
226071³ = 11554058606155911 ; 11554+058606+155911 = 226071
279720³ = 21886209834048000 ; 21886+209834+048000 = 279720
313390³ = 30779063611219000 ; 30779+063611+219000 = 313390
324675³ = 34225243575046875 ; 34225+243575+046875 = 324675
329967³ = 35926219978074063 ; 35926+219978+074063 = 329967
346060³ = 41443288617016000 ; 41443+288617+016000 = 346060
368928³ = 50214003962314752 ; 50214+003962+314752 = 368928
395604³ = 61913024827308864 ; 61913+024827+308864 = 395604
422577³ = 75460133084214033 ; 75460+133084+214033 = 422577
427868³ = 78330233506116032 ; 78330+233506+116032 = 427868
461539³ = 98316229404133819 ; 98316+229404+133819 = 461539
472823³ = 105705061351305767 ; 105705+061351+305767 = 472823
478115³ = 109294197946170875 ; 109294+197946+170875 = 478115
488214³ = 116367227503144344 ; 116367+227503+144344 = 488214
494208³ = 120706126590246912 ; 120706+126590+246912 = 494208
495208³ = 121440334856038912 ; 121440+334856+038912 = 495208
499500³ = 124625374875000000 ; 124625+374875+000000 = 499500
500500³ = 125375375125000000 ; 125375+375125+000000 = 500500
517076³ = 138249363851014976 ; 138249+363851+014976 = 517076
533170³ = 151564368606013000 ; 151564+368606+013000 = 533170
543752³ = 160769107975275008 ; 160769+107975+275008 = 543752
559846³ = 175471156639227736 ; 175471+156639+227736 = 559846
565137³ = 180493358291026353 ; 180493+358291+026353 = 565137
598807³ = 214714120150263943 ; 214714+120150+263943 = 598807
664741³ = 293736149984221021 ; 293736+149984+221021 = 664741
670032³ = 300806096458272768 ; 300806+096458+272768 = 670032
720279³ = 373682068958277639 ; 373682+068958+277639 = 720279
757835³ = 435235164725157875 ; 435235+164725+157875 = 757835
791505³ = 495862183018112625 ; 495862+183018+112625 = 791505
807598³ = 526727149679131192 ; 526727+149679+131192 = 807598
825175³ = 561873028927234375 ; 561873+028927+234375 = 825175
829466³ = 570684092086166696 ; 570684+092086+166696 = 829466
856142³ = 627534213320015288 ; 627534+213320+015288 = 856142
966329³ = 902350034690029289 ; 902350+034690+029289 = 966329
973323³ = 922085001971049267 ; 922085+001971+049267 = 973323
4444443³ = 87791409602222606307 ; 877914+0960222+2606307 = 4444443
4927940³ = 119673015472102184000 ; 1196730+1547210+2184000 = 4927940
5072058³ = 130482609481202819112 ; 1304826+0948120+2819112 = 5072058
5555555³ = 171467712620032578875 ; 1714677+1262003+2578875 = 5555555
5555556³ = 171467805212623319616 ; 1714678+0521262+3319616 = 5555556
5699673³ = 185161129138450934217 ; 1851611+2913845+0934217 = 5699673
6183170³ = 236392428062461013000 ; 2363924+2806246+1013000 = 6183170
8888887³ = 702331513854690480103 ; 7023315+1385469+0480103 = 8888887
11273318³ = 1432700041661610713432 ; 143270+00416616+10713432 = 11273318
13793570³ = 2624400123813012293000 ; 262440+01238130+12293000 = 13793570
17090613³ = 4991980961501806976397 ; 499198+09615018+06976397 = 17090613
21803275³ = 10364901934491001421875 ; 1036490+19344910+01421875 = 21803275
22293325³ = 11079611748223903703125 ; 1107961+17482239+03703125 = 22293325
24752475³ = 15165470606405317171875 ; 1516547+06064053+17171875 = 24752475
25242525³ = 16084160018098423453125 ; 1608416+00180984+23453125 = 25242525
25252524³ = 16103281230837211333824 ; 1610328+12308372+11333824 = 25252524
27272728³ = 20285501247182612772352 ; 2028550+12471826+12772352 = 27272728
27282727³ = 20307821247736212774583 ; 2030782+12477362+12774583 = 27282727
28201724³ = 22429881232731213631424 ; 2242988+12327312+13631424 = 28201724
30731977³ = 29024951215764915671833 ; 2902495+12157649+15671833 = 30731977
33404436³ = 37274551844312511233856 ; 3727455+18443125+11233856 = 33404436
36363635³ = 48084141848234613072875 ; 4808414+18482346+13072875 = 36363635
38383839³ = 56551642990995602818719 ; 5655164+29909956+02818719 = 38383839
38546045³ = 57271620577775827041125 ; 5727162+05777758+27041125 = 38546045
38883889³ = 58790761308044419924369 ; 5879076+13080444+19924369 = 38883889
39046095³ = 59529580178576231307375 ; 5952958+01785762+31307375 = 39046095
39546145³ = 61846120636290826998625 ; 6184612+06362908+26998625 = 39546145
41843088³ = 73260720125954433257472 ; 7326072+01259544+33257472 = 41843088
44025497³ = 85332172405380711438473 ; 8533217+24053807+11438473 = 44025497
44363340³ = 87311752792816507704000 ; 8731175+27928165+07704000 = 44363340
44525547³ = 88272980485592630842323 ; 8827298+04855926+30842323 = 44525547
45025597³ = 91280590244636533451173 ; 9128059+02446365+33451173 = 45025597
47045800³ = 104126812463311912000000 ; 10412681+24633119+12000000 = 47045800
49842793³ = 123824650720907130251257 ; 12382465+07209071+30251257 = 49842793
49995000³ = 124962503749875000000000 ; 12496250+37498750+00000000 = 49995000
50005000³ = 125037503750125000000000 ; 12503750+37501250+00000000 = 50005000
50657256³ = 129994501742459020233216 ; 12999450+17424590+20233216 = 50657256
55474451³ = 170717891744681120955851 ; 17071789+17446811+20955851 = 55474451
55474452³ = 170717900976925428633408 ; 17071790+09769254+28633408 = 55474452
55484452³ = 170810240065002037753408 ; 17081024+00650020+37753408 = 55484452
56136708³ = 176905291183926726606912 ; 17690529+11839267+26606912 = 56136708
58156911³ = 196699833740089701086031 ; 19669983+37400897+01086031 = 58156911
58656961³ = 201817431964353718831681 ; 20181743+19643537+18831681 = 58656961
60453854³ = 220938793708011101279864 ; 22093879+37080111+01279864 = 60453854
61453954³ = 232086293635066101894664 ; 23208629+36350661+01894664 = 61453954
63636364³ = 257700981126972226596544 ; 25770098+11269722+26596544 = 63636364
63798570³ = 259676610203790935793000 ; 25967661+02037909+35793000 = 63798570
64298620³ = 265830590578756131928000 ; 26583059+05787561+31928000 = 64298620
66747770³ = 297378990157687135433000 ; 29737899+01576871+35433000 = 66747770
69278022³ = 332496010036177335666648 ; 33249601+00361773+35666648 = 69278022
69768072³ = 339601941949863016309248 ; 33960194+19498630+16309248 = 69768072
69778072³ = 339747990397402531829248 ; 33974799+03974025+31829248 = 69778072
72227222³ = 376792920320288231345048 ; 37679292+03202882+31345048 = 72227222
72717272³ = 384514510655817327707648 ; 38451451+06558173+27707648 = 72717272
72727271³ = 384673150653644527723511 ; 38467315+06536445+27723511 = 72727271
74747475³ = 417627972518780307796875 ; 41762797+25187803+07796875 = 74747475
74909681³ = 420352701580617017068241 ; 42035270+15806170+17068241 = 74909681
75409731³ = 428827052296213509564891 ; 42882705+22962135+09564891 = 75409731
77706674³ = 469218322003481810750024 ; 46921832+20034818+10750024 = 77706674
77767777³ = 470326072320873707526433 ; 47032607+23208737+07526433 = 77767777
77777776³ = 470507512318244907544576 ; 47050751+23182449+07544576 = 77777776
80726976³ = 526085161987628408242176 ; 52608516+19876284+08242176 = 80726976
80726977³ = 526085181538162612736833 ; 52608518+15381626+12736833 = 80726977
80889183³ = 529262771767141910291487 ; 52926277+17671419+10291487 = 80889183
88878888³ = 702094930154632317123072 ; 70209493+01546323+17123072 = 88878888
88888888³ = 702331940521262213443072 ; 70233194+05212622+13443072 = 88888888
93868290³ = 827097521036953800789000 ; 82709752+10369538+00789000 = 93868290
197864531³ = 7746470118484770071633291 ; 7746470+118484770+071633291 = 197864531
332999667³ = 36925926221999778074073963 ; 36925926+221999778+074073963 = 332999667
667000332³ = 296741406109664558260594368 ; 296741406+109664558+260594368 = 667000332
765432098³ = 448456177233250729083725192 ; 448456177+233250729+083725192 = 765432098
1111111110³ = 1371742108367626890260631000 ; 13717421+0836762689+0260631000 = 1111111110
1153093348³ = 1533180900156013471122160192 ; 15331809+0015601347+1122160192 = 1153093348
1355013550³ = 2487888510162596650313875000 ; 24878885+1016259665+0313875000 = 1355013550
1396995787³ = 2726373106688236530700908403 ; 27263731+0668823653+0700908403 = 1396995787
2020202019³ = 8244881202309803481706772859 ; 82448812+0230980348+1706772859 = 2020202019
2020202020³ = 8244881214553452080482408000 ; 82448812+1455345208+0482408000 = 2020202020
2112639308³ = 9429226502268409311791506112 ; 94292265+0226840931+1791506112 = 2112639308
2306086696³ = 12263851721309666430052481536 ; 0122638517+2130966643+0052481536 = 2306086696
2685390489³ = 19365215722482381630243500169 ; 0193652157+2248238163+0243500169 = 2685390489
2887410691³ = 24072748607741138341872569371 ; 0240727486+0774113834+1872569371 = 2887410691
(checked all bases through 2969000000)

Table of expansions of order-4 Kaprekar numbers:

1⁴ = 1 ; 0+0+0+1 = 1
7⁴ = 2401 ; 2+4+0+1 = 7
45⁴ = 4100625 ; 4+10+06+25 = 45
55⁴ = 9150625 ; 9+15+06+25 = 55
67⁴ = 20151121 ; 20+15+11+21 = 67
100⁴ = 100000000 ; 0+100+000+000 = 100
433⁴ = 35152125121 ; 35+152+125+121 = 433
4950⁴ = 600372506250000 ; 600+3725+0625+0000 = 4950
5050⁴ = 650377506250000 ; 650+3775+0625+0000 = 5050
38212⁴ = 2132058301111419136 ; 2132+05830+11114+19136 = 38212
65068⁴ = 17925440300173701376 ; 17925+44030+01737+01376 = 65068
190576⁴ = 1319085144029937074176 ; 1319+085144+029937+074176 = 190576
295075⁴ = 7581055275091594140625 ; 7581+055275+091594+140625 = 295075
299035⁴ = 7996281784008630000625 ; 7996+281784+008630+000625 = 299035
310024⁴ = 9238070268138742091776 ; 9238+070268+138742+091776 = 310024
336700⁴ = 12852051748272100000000 ; 12852+051748+272100+000000 = 336700
343333⁴ = 13895116532078585134321 ; 13895+116532+078585+134321 = 343333
394615⁴ = 24249029288190453150625 ; 24249+029288+190453+150625 = 394615
414558⁴ = 29535287434085093012496 ; 29535+287434+085093+012496 = 414558
433566⁴ = 35336283531075963038736 ; 35336+283531+075963+038736 = 433566
448228⁴ = 40364161075126533120256 ; 40364+161075+126533+120256 = 448228
450550⁴ = 41207092837066506250000 ; 41207+092837+066506+250000 = 450550
467236⁴ = 47659028925138236252416 ; 47659+028925+138236+252416 = 467236
475497⁴ = 51120032934245362146081 ; 51120+032934+245362+146081 = 475497
476191⁴ = 51419130918046493247361 ; 51419+130918+046493+247361 = 476191
486486⁴ = 56012039572176486214416 ; 56012+039572+176486+214416 = 486486
499500⁴ = 62250374750062500000000 ; 62250+374750+062500+000000 = 499500
500500⁴ = 62750375250062500000000 ; 62750+375250+062500+000000 = 500500
523513⁴ = 75112096258305582046561 ; 75112+096258+305582+046561 = 523513
534898⁴ = 81862291297038523123216 ; 81862+291297+038523+123216 = 534898
549550⁴ = 91207142337066006250000 ; 91207+142337+066006+250000 = 549550
599743⁴ = 129378094625105339270401 ; 129378+094625+105339+270401 = 599743
622414⁴ = 150078130395081125260816 ; 150078+130395+081125+260816 = 622414
628408⁴ = 155943336115129054007296 ; 155943+336115+129054+007296 = 628408
647362⁴ = 175626000494277306193936 ; 175626+000494+277306+193936 = 647362
652015⁴ = 180730041458379202050625 ; 180730+041458+379202+050625 = 652015
671671⁴ = 203529043026143035282081 ; 203529+043026+143035+282081 = 671671
677755⁴ = 211004112934353192000625 ; 211004+112934+353192+000625 = 677755
705331⁴ = 247498110530114582232721 ; 247498+110530+114582+232721 = 705331
731368⁴ = 286117102194115281227776 ; 286117+102194+115281+227776 = 731368
734932⁴ = 291735063830077991301376 ; 291735+063830+077991+301376 = 734932
765963⁴ = 344216089001250585082161 ; 344216+089001+250585+082161 = 765963
772200⁴ = 355565151035265600000000 ; 355565+151035+265600+000000 = 772200
803539⁴ = 416896108155199047079441 ; 416896+108155+199047+079441 = 803539
820612⁴ = 453473020676298527047936 ; 453473+020676+298527+047936 = 820612
826606⁴ = 466868113631102411143696 ; 466868+113631+102411+143696 = 826606
849520⁴ = 520828128408040284160000 ; 520828+128408+040284+160000 = 849520
934066⁴ = 761220115590026520030736 ; 761220+115590+026520+030736 = 934066
25280200⁴ = 408434126221501224961600000000 ; 408434+12622150+12249616+00000000 = 25280200
33333334⁴ = 1234568000000002962963002469136 ; 1234568+00000000+29629630+02469136 = 33333334
44525548⁴ = 3930403094533271344940217692416 ; 3930403+09453327+13449402+17692416 = 44525548
49995000⁴ = 6247500374975000625000000000000 ; 6247500+37497500+06250000+00000000 = 49995000
50005000⁴ = 6252500375025000625000000000000 ; 6252500+37502500+06250000+00000000 = 50005000
55474452⁴ = 9470482003285202798303417692416 ; 9470482+00328520+27983034+17692416 = 55474452
58585858⁴ = 11780702219802231731003707514896 ; 11780702+21980223+17310037+07514896 = 58585858
61643836⁴ = 14439715129046200264868531650816 ; 14439715+12904620+02648685+31650816 = 61643836
77696674⁴ = 36442630072556942449817409500176 ; 36442630+07255694+24498174+09500176 = 77696674
332999667⁴ = 12296321135592494172740531012370321 ; 12296321+135592494+172740531+012370321 = 332999667
334333333⁴ = 12494495111665332074519185135654321 ; 12494495+111665332+074519185+135654321 = 334333333
811144477⁴ = 432905259131466615113518762133253841 ; 432905259+131466615+113518762+133253841 = 811144477
3577235773⁴ = 163753375309125813101733384260148885841 ; 163753375+3091258131+0173338426+0148885841 = 3577235773
(checked all bases through 3611622599)

Table of expansions of order-5 Kaprekar numbers:

1⁵ = 1 ; 0+0+0+0+1 = 1
10⁵ = 100000 ; 0+00+10+00+00 = 10
1000⁵ = 1000000000000000 ; 0+1000+0000+0000+0000 = 1000
7776⁵ = 28430288029929701376 ; 2843+0288+0299+2970+1376 = 7776
27100⁵ = 14616603103510000000000 ; 146+16603+10351+00000+00000 = 27100
73440⁵ = 2136305413264402022400000 ; 21363+05413+26440+20224+00000 = 73440
95120⁵ = 7786803368050520883200000 ; 77868+03368+05052+08832+00000 = 95120
500499⁵ = 31406249062033782183750002499 ; 31406+249062+033782+183750+002499 = 500499
505791⁵ = 33102095432189229078077109951 ; 33102+095432+189229+078077+109951 = 505791
540539⁵ = 46146117682034200315812026699 ; 46146+117682+034200+315812+026699 = 540539
598697⁵ = 76919315312001453031756173257 ; 76919+315312+001453+031756+173257 = 598697
665335⁵ = 130377260305079059111219084375 ; 130377+260305+079059+111219+084375 = 665335
697598⁵ = 165206120951064038339435007968 ; 165206+120951+064038+339435+007968 = 697598
732347⁵ = 210661190534180357124288026507 ; 210661+190534+180357+124288+026507 = 732347
7607610⁵ = 25482451314582286738507773980100000 ; 2548245+1314582+2867385+0777398+0100000 = 7607610
37944478⁵ = 78658002460354141239901176718608806368 ; 786580+02460354+14123990+11767186+08806368 = 37944478
(checked all bases through 44275337)

Some other sequences are discussed here.

Men Core Values

© 1996-2008 Robert P. Munafo. Email the author

This work is licensed under a Creative Commons Attribution 2.5 License.

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