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Kaprekar numbers
A number  $n$  whose square can be partitioned into two parts whose sum is  $n$  itself is called Kaprekar number.

For example,  $45$  is a Kaprekar number, because  $45^2 = \underline{20}\,\,\underline{25}$  and  $20+25=45$.

Note that the second part can start with zero:  $5292^2 = \underline{28}\,\,\underline{005264}$  and  $28+5264=5292$.

D. E. Iannucci has proved that the Kaprekar numbers whose second part consists of  $n$  digits are in one-to-one correspondence with the unitary divisors of  $10^n-1$.

The first Kaprekar numbers are 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 more terms

Kaprekar numbers can also be... (you may click on names or numbers and on + to get more values)

aban 45 55 99 297 + 667000333 abundant 2728 4950 5292 7272 + 49995000 Achilles 5292 alternating 45 703 5050 5292 + 909090909 amenable 45 297 2728 5292 + 909090909 apocalyptic 2223 2728 5050 7272 + 22222 arithmetic 45 55 99 297 + 9999999 binomial 45 55 703 4950 + 500000500000 brilliant 703 5072059 c.nonagonal 55 703 5050 500500 + 50000005000000 cake 17344 canyon 703 961038 Carmichael 670033 congruent 45 55 703 999 + 9999999 Cunningham 99 999 7777 9999 + 99999999999999 Curzon 4950 77778 95121 148149 + 961038 cyclic 703 4879 7777 181819 + 5479453 d-powerful 2223 4950 356643 9372385 de Polignac 94520547 decagonal 297 466830 deceptive 703 7777 390313 670033 + 1111111111 deficient 45 55 99 297 + 9999999 dig.balanced 99 500500 643357 648648 + 55474452 Duffinian 55 999 7777 95121 + 5072059 economical 17344 148149 emirpimes 181819 5072059 equidigital 17344 148149 esthetic 45 evil 45 99 297 703 + 909090909 Fibonacci 55 gapful 297 5050 7272 7777 + 74074074075 happy 17344 208495 390313 499500 + 8161912 Harshad 45 999 2223 4950 + 8888888889 heptagonal 55 356643 670033 hexagonal 45 703 4950 499500 + 49999995000000 hoax 851851 25252525 Hogben 703 iban 703 2223 7272 7777 + 22222 idoneal 45 inconsummate 95121 142857 148149 538461 + 961038 interprime 45 99 77778 95121 + 86358636 junction 22222 95121 466830 681318 + 86358636 Lehmer 703 670033 lucky 99 297 9999 329967 + 9372385 metadrome 45 modest 999 7777 9999 22222 + 1111111111 Moran 45 999 5555556 mountain 297 4950 38962 nialpdrome 55 99 999 7777 + 999999999999999 nonagonal 71428071429 nude 55 99 999 7777 + 432432432 oban 55 99 703 999 odious 55 2223 2728 4879 + 999999999 palindromic 55 99 999 7777 + 999999999999999 panconsummate 45 pandigital 99 5479453 partition 297 pernicious 55 2223 2728 4879 + 5072059 plaindrome 45 55 99 999 + 999999999999999 Poulet 670033 powerful 5292 practical 2728 4950 5292 7272 + 8161912 prim.abundant 999999 13641364 pronic 82656 pseudoperfect 2728 4950 5292 7272 + 999999 repdigit 55 99 999 7777 + 999999999999999 repunit 703 1111111111 self 703 38962 142857 318682 + 74747475 self-describing 88888888 semiprime 55 703 181819 857143 5072059 Smith 681318 851851 49995000 55636659 sphenic 4879 7777 22222 5479453 + 80726977 straight-line 999 7777 9999 22222 + 999999999999999 strobogrammatic 88888888 1111111111 super Niven 5050 500500 50005000 5000050000 super-d 181819 818181 tau 5292 7272 82656 5555556 11111112 triangular 45 55 703 4950 + 50000005000000 trimorphic 99 999 9999 99999 + 999999999999999 uban 45 55 99 50000005000000 Ulam 99 142857 187110 208495 + 9999999 undulating 5050 7272 818181 25252525 + 85858585858585 unprimeable 7272 38962 466830 791505 + 9372385 untouchable 5292 7272 77778 82656 + 500500 upside-down 55 55555555555 wasteful 45 55 99 297 + 9999999 Woodall 99999999999 Zuckerman 11111112 1111111111 Zumkeller 2728 4950 5292 7272 + 82656 zygodrome 55 99 999 7777 + 999999999999999