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On Some Formulas for Kaprekar Constants

Department of Information Systems Science, Soka University, Tokyo 192-8577, Japan
Author to whom correspondence should be addressed.
Symmetry 2019, 11(7), 885;
Received: 3 June 2019 / Revised: 1 July 2019 / Accepted: 2 July 2019 / Published: 5 July 2019
(This article belongs to the Special Issue Number Theory and Symmetry)
PDF [330 KB, uploaded 12 July 2019]


Let b 2 and n 2 be integers. For a b-adic n-digit integer x, let A (resp. B) be the b-adic n-digit integer obtained by rearranging the numbers of all digits of x in descending (resp. ascending) order. Then, we define the Kaprekar transformation T ( b , n ) ( x ) : = A B . If T ( b , n ) ( x ) = x , then x is called a b-adic n-digit Kaprekar constant. Moreover, we say that a b-adic n-digit Kaprekar constant x is regular when the numbers of all digits of x are distinct. In this article, we obtain some formulas for regular and non-regular Kaprekar constants, respectively. As an application of these formulas, we then see that for any integer b 2 , the number of b-adic odd-digit regular Kaprekar constants is greater than or equal to the number of all non-trivial divisors of b. Kaprekar constants have the symmetric property that they are fixed points for recursive number theoretical functions T ( b , n ) . View Full-Text
Keywords: Kaprekar constants; Kaprekar transformation; fixed points for recursive functions Kaprekar constants; Kaprekar transformation; fixed points for recursive functions
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Yamagami, A.; Matsui, Y. On Some Formulas for Kaprekar Constants. Symmetry 2019, 11, 885.

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