A number n whose square can be partitioned into two parts whose sum is n itself. more
The Kaprekar numbers up to 10
15 :
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 |
2 | 183 | 202 | 2 |
3 | 200 | 185 | 0 | 3 |
4 | 114 | 95 | 69 | 107 | 4 |
5 | 80 | 72 | 74 | 72 | 87 | 5 |
6 | 93 | 95 | 0 | 107 | 90 | 0 | 6 |
7 | 110 | 112 | 33 | 31 | 37 | 38 | 24 | 7 |
8 | 53 | 48 | 35 | 47 | 61 | 47 | 34 | 60 | 8 |
9 | 200 | 185 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 9 |
10 | 46 | 38 | 36 | 38 | 33 | 34 | 34 | 38 | 34 | 54 | 10 |
11 | 148 | 144 | 8 | 13 | 7 | 12 | 7 | 13 | 8 | 16 | 9 |
A pictorial representation of the table above
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.