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Kaprekar number




Take a positive whole number n that has d number of digits. Take the square n and separate the result into two pieces: a right-hand piece that has d digits and a left-hand piece that has either d or d-1 digits. Add these two pieces together. If the result is n, then n is a Kaprekar number. Examples are 9 (92 = 81, 8 + 1 = 9), 45 (452 = 2025, 20 + 25 = 45), and 297 (2972 = 88209, 88 + 209 = 297).

The first 20 Kaprekar numbers according to this definition are 1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4950, 5050, 7272, 7777, 9999, 17344, 22222, 77778, 82656, 95121, and 99999.

Kaprekar numbers can also be defined by higher powers. For example, 453 = 91125, and 9 + 11 + 25 = 45. The first ten numbers with this property are: 1, 8, 10, 45, 297, 2322, 2728, 4445, 4544, and 4949. For fourth powers, the sequence begins 1, 7, 45, 55, 67, (100), 433, 4950, 5050, 38212, 65068. Notice that 45 is a Kaprekar number for second, third, and fourth powers (454 = 4100625, and 4 + 10 + 06 + 25 = 45) – the only number in all three Kaprekar sequences, up to at least 400,000.


Related entry

   • unique number


Related category

   • TYPES OF NUMBERS