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# On Some Formulas for Kaprekar Constants

Department of Information Systems Science, Soka University, Tokyo 192-8577, Japan
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Author to whom correspondence should be addressed.
Symmetry 2019, 11(7), 885; https://doi.org/10.3390/sym11070885
Received: 3 June 2019 / Revised: 1 July 2019 / Accepted: 2 July 2019 / Published: 5 July 2019
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PDF [330 KB, uploaded 12 July 2019]

# Abstract

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
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).

MDPI and ACS Style

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

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