Kaprekar number

In mathematics, a Kaprekar number is a number that, in a given base, when squared, can be split into two numbers with the same number of digits as the original number which add up to the original number again. For example, the 3-digit number 703 is a Kaprekar number, because 703² = 494209, which can be split into 494 and 209, and 494 + 209 = 703.

The first few Kaprekar numbers in base 10 are (sequence A006886 in OEIS):

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

In binary, all perfect numbers are Kaprekar numbers.

The Kaprekar numbers are named after D. R. Kaprekar.

References

• D. R. Kaprekar, On Kaprekar numbers, J. Rec. Math., 13 (1980–1981), 81–82.
• M. Charosh, Some Applications of Casting Out 999...'s, Journal of Recreational Mathematics 14, 1981–82, pp. 111–118
• Douglas E. Iannucci, The Kaprekar Numbers, Journal of Integer Sequences, Vol. 3 (2000), http://www.math.uwaterloo.ca/JIS/VOL3/iann2a.html