Kaprekar Sequences
Wells' 1986 book [1] defines Kaprekar number 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. The name "Kaprekar triple" was given by Wells ([1] in the entry for 297) and is also used by Iannucci [3] There are also solutions for higher powers.
Douglas E. Iannucci [2] explains how the Kaprekar numbers (for squares) can be derived from the prime factorization of 10D-1. He also shows that if N is a Kaprekar number of D digits, then 10D-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: 4879 is paired with 95121 (95121+04879 = 105) so if we disallow 4879, 95121 would be without a mate.
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, Q≥0 and 0≤R<10D
such that N2=Q×10D+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). Sloane's A045913 uses this kind of definition. A006886 is similar but excludes powers of 10.
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 pair-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, Q≥0 and 0≤R<10D
such that 10D-1 ≤ N < 10D,
N2=Q×10D+R and Q+R=N.
For clarity 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.
Sloane's catalog includes A006887, a version for 3rd powers that is self-contradictory: it includes the solution 271003 = 19902511000000 ; 01990+25110+00000 = 27100, but does not include 103 = 1000 ; 00+10+00 = 10. Presumably the reason is that 0's are allowed in the 2rd or 3rd digit-group, but not in the first.
Not all of the variations of each Kaprekar sequence are included in Sloane's catalog, but with some work one can be used to derive another. I consider the "natural" version of definition C to be the most useful, even if my notion of "naturalness" is rejected:
- The sequences that omit powers of 10 can very easily be derived by omitting powers of 10 from the natural version.
- For the N2 case, solutions like 048792=00238 04641 can be deduced by applying Kaprekar's symmetry that the numbers add in pairs to form a power of 10.
Therefore, the natural versions contain all the information one would need to get the other two versions, and for that reason I will give only the natural versions here.
Here are the first four "natural" Kaprekar number sequences:
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, ... (Sloane's A053816)
For 3rd 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, ...
(Sloane's A060809, currently described as "erroneous")
For 4th 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, ...
(Sloane's A171493)
For 5th powers: 1, 10, 1000, 7776, 27100, 73440, 95120, 500499, 505791, 540539, 598697, 665335, 697598, 732347, 7607610, 37944478, 46945205, 54995500, 55216205, 56607166, ...
(Sloane's A171500)
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
(No other solutions below 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
(No other solutions below 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
(No other solutions below 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
(No other solutions below 44275337)
Some other sequences are discussed here.
References
[1] Wells, David, The Penguin Dictionary of Curious and Interesting Numbers. (Penguin Books, 1986) (revised 1997) ISBN 0-14-026149-4
[2] Douglas E. Iannucci, The Kaprekar Numbers. Journal of Integer Sequences, 3 (2000), Article 00.1.2 text online
[3] Douglas E. Iannucci, The Kaprekar Numbers. Journal of Integer Sequences, 3 (2000), Article 00.1.2 text online
Wikipedia, Kaprekar number, encyclopedia page.
This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2011 May 10.
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