[go: up one dir, main page]

Search a number
Kaprekar numbers
A number n whose square can be partitioned into two parts whose sum is n itself. more

The Kaprekar numbers up to 1015 :
1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777, 9999, 17344, 22222, 38962, 77778, 82656, 95121, 99999, 142857, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170, 538461, 609687, 627615, 643357, 648648, 670033, 681318, 791505, 812890, 818181, 851851, 857143, 961038, 994708, 999999, 4444444, 4927941, 5072059, 5479453, 5555556, 8161912, 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, 243902440, 332999667, 432432432, 567567568, 665188470, 667000333, 765432099, 867208672, 909090909, 999999999, 1111111111, 1776299581, 2020202020, 2646002646, 3846956652, 3888938889, 4090859091, 4132841328, 4756047561, 4798029798, 4958067763, 4999950000, 5000050000, 5041932237, 5201970202, 5243952439, 5867158672, 5909140909, 6111061111, 6153043348, 7359343993, 7979797980, 8223700419, 8888888889, 8975672343, 9090909091, 9132791328, 9334811530, 9756097560, 9999999999, 19481019481, 20305787796, 24138656649, 44444444445, 55555555555, 65098401732, 71428071429, 74074074075, 74761738129, 75861343351, 79694212204, 81433418067, 81933418567, 90909090910, 93555093555, 98268434902, 99999999999, 104247104248, 148835812203, 156007492641, 159341159341, 164983164984, 165670829038, 168316831684, 172342508976, 175676175676, 178321178322, 195156195157, 218400870420, 230769230770, 233415233415, 238128574762, 239744903112, 242390905758, 249750249750, 250250250250, 259225922593, 266585266585, 269230269231, 272563935931, 304843304844, 312202648836, 321678321679, 324324324324, 329037665671, 331683668317, 333299996667, 335016335017, 340659340659, 346638010005, 363473026840, 395752395753, 399086062453, 403111739745, 405757742391, 406445406445, 409090409091, 422592759226, 425925425926, 428571428572, 435930772564, 437547100914, 473160136527, 480519480519, 489995153362, 496666833300, 497354497354, 499999500000, 500000500000, 502645502646, 503333166700, 510004846638, 519480519481, 526839863473, 562452899086, 564069227436, 571428571428, 574074574074, 577407240774, 590909590909, 593554593555, 594242257609, 596888260255, 600913937547, 604247604247, 636526973160, 653361989995, 659340659341, 664983664983, 666700003333, 668316331683, 670962334329, 675675675676, 678321678321, 687797351164, 695156695156, 727436064069, 730769730769, 733414733415, 740774077407, 749749749750, 750249750250, 757609094242, 760255096888, 761871425238, 766584766585, 769230769230, 804843804843, 821678821678, 824323824324, 827657491024, 831683168316, 834329170962, 835016835016, 840658840659, 843992507359, 851164187797, 895752895752, 901731565098, 906444906445, 909090909090, 918066581433, 918566581933, 925238261871, 925925925925, 928571928571, 934901598268, 980518980519, 991024327657, 992640656007, 997353997354, 999999999999, 1174004192872, 1392405063292, 2003821351803, 2222222222223, 2902798571724, 3138070313808, 3396226415094, 3614627285514, 6385372714486, 6603773584906, 7777777777777, 7996178648197, 8585853585859, 8607594936708, 8825995807128, 9781599129580, 9999999999999, 13636368636364, 14141414141415, 16774438950171, 22222222222222, 24639702463971, 25360292536030, 27777782777778, 36363636363637, 38276071100334, 38996661172393, 39501706677444, 46861924686193, 49999995000000, 50000005000000, 53138075313807, 57813911472448, 60498293322556, 60716710044172, 61003338827607, 61723928899666, 63636363636363, 68315548020345, 71167942135684, 71218346592069, 72222217222222, 74070740707408, 74639707463970, 75360297536029, 77777777777778, 83225561049829, 85858585858585, 86363631363636, 91414146414141, 96861929686192, 99999999999999, 126129459492793, 129032258064517, 188040912667191, 190943711238915, 200200200200200, 203102998771924, 220564571237110, 223467369808834, 256356460687536, 259259259259260, 273972602739727, 317073170731707, 319975969303431, 330427201394943, 333329999966667, 346942379940013, 349596829301626, 352499627873350, 367738627206798, 385388718752052, 388241112867391, 388291517323776, 391143911439115, 408095604080956, 459459459459459, 462362258031183, 479823830496369, 482726629068093, 494402381414613, 517273370931907, 520176169503631, 537637741968817, 540540540540541, 555555605555556, 608856088560885, 611708482676224, 611758887132609, 614611281247948, 620914982679739, 641711229946524, 647500372126650, 650403170698374, 666670000033333, 669572798605057, 680024030696569, 682068206820682, 682926829268293, 740740740740740, 743643539312464, 768374984154339, 776532630191166, 779435428762890, 796897001228076, 799799799799800, 809056288761085, 811959087332809, 829528208295282, 870967741935483, 873870540507207, 925929259292592, 928781653407931, 928832057864316, 931684451979655, 939283289955828, 942186088527552, 963651109636512, 988653609886537, 997097201428276, 999999999999999.

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 385 values, from 1 to 999999999999999).

n\r 0  1 
2183202 2 
32001850 3 
41149569107 4 
58072747287 5 
693950107900 6 
71101123331373824 7 
85348354761473460 8 
92001850000000 9 
1046383638333434383454 10 
111481448137127138169

A pictorial representation of the table above
motab
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.