Kaprekar numbers

A number

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

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

A graph displaying how many Kaprekar numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links Mathworld, Kaprekar Number
Wikipedia, Kaprekar number
OEIS, Sequence A006886

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

aban 45 55 99 + 999 667000333 abundant 2728 4950 5292 + 36363636 49995000 Achilles 5292 alternating 45 703 5050 + 36363636 909090909 amenable 45 297 2728 + 867208672 909090909 apocalyptic 2223 2728 5050 + 17344 22222 arithmetic 45 55 99 + 9372385 9999999 binomial 45 55 703 + 499999500000 500000500000 brilliant 703 5072059 c.nonagonal 55 703 5050 + 500000500000 50000005000000 cake 17344 Carmichael 670033 congruent 45 55 703 + 9372385 9999999 Cunningham 99 999 7777 + 9999999999999 99999999999999 Curzon 4950 77778 95121 + 818181 961038 cyclic 703 4879 7777 + 5072059 5479453 d-powerful 2223 4950 356643 9372385 de Polignac 94520547 decagonal 297 466830 deceptive 703 7777 390313 + 667000333 1111111111 deficient 45 55 99 + 9372385 9999999 dig.balanced 99 500500 643357 + 5479453 55474452 Duffinian 55 999 7777 + 857143 5072059 economical 17344 148149 emirpimes 181819 5072059 equidigital 17344 148149 esthetic 45 evil 45 99 297 + 867208672 909090909 Fibonacci 55 gapful 297 5050 7272 + 44444444445 74074074075 happy 17344 208495 390313 + 994708 8161912 Harshad 45 999 2223 + 5000050000 8888888889 heptagonal 55 356643 670033 hexagonal 45 703 4950 + 499999500000 49999995000000 hoax 851851 25252525 Hogben 703 iban 703 2223 7272 + 17344 22222 idoneal 45 inconsummate 95121 142857 148149 + 648648 961038 interprime 45 99 77778 + 63636364 86358636 junction 22222 95121 466830 + 83409436 86358636 Lehmer 703 670033 lucky 99 297 9999 + 461539 9372385 metadrome 45 modest 999 7777 9999 + 999999999 1111111111 Moran 45 999 5555556 nialpdrome 55 99 999 + 99999999999999 999999999999999 nonagonal 71428071429 nude 55 99 999 + 99999999 432432432 oban 55 99 703 999 odious 55 2223 2728 + 667000333 999999999 palindromic 55 99 999 + 99999999999999 999999999999999 panconsummate 45 pandigital 99 5479453 partition 297 pernicious 55 2223 2728 + 4927941 5072059 plaindrome 45 55 99 + 99999999999999 999999999999999 Poulet 670033 powerful 5292 practical 2728 4950 5292 + 961038 8161912 prim.abundant 999999 13641364 pronic 82656 pseudoperfect 2728 4950 5292 + 994708 999999 repdigit 55 99 999 + 99999999999999 999999999999999 repunit 703 1111111111 self 703 38962 142857 + 36363636 74747475 self-describing 88888888 semiprime 55 703 181819 857143 5072059 Smith 681318 851851 49995000 55636659 sphenic 4879 7777 22222 + 19773073 80726977 straight-line 999 7777 9999 + 99999999999999 999999999999999 strobogrammatic 88888888 1111111111 super Niven 5050 500500 50005000 5000050000 super-d 181819 818181 tau 5292 7272 82656 5555556 11111112 triangular 45 55 703 + 49999995000000 50000005000000 trimorphic 99 999 9999 + 99999999999999 999999999999999 uban 45 55 99 50000005000000 Ulam 99 142857 187110 + 609687 9999999 undulating 5050 7272 818181 + 63636363636363 85858585858585 unprimeable 7272 38962 466830 + 5555556 9372385 untouchable 5292 7272 77778 + 499500 500500 upside-down 55 55555555555 wasteful 45 55 99 + 9372385 9999999 Woodall 99999999999 Zuckerman 11111112 1111111111 Zumkeller 2728 4950 5292 + 77778 82656 zygodrome 55 99 999 + 99999999999999 999999999999999