On Some Formulas for Kaprekar Constants
Department of Information Systems Science, Soka University, Tokyo 192-8577, Japan
Author to whom correspondence should be addressed.
Received: 3 June 2019 / Revised: 1 July 2019 / Accepted: 2 July 2019 / Published: 5 July 2019
PDF [330 KB, uploaded 12 July 2019]
be integers. For a b
-digit integer x
, let A
) be the b
-digit integer obtained by rearranging the numbers of all digits of x
in descending (resp. ascending) order. Then, we define the Kaprekar transformation
, then x
is called a b
-digit Kaprekar constant
. Moreover, we say that a b
-digit Kaprekar constant x
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
, 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
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MDPI and ACS Style
Yamagami, A.; Matsui, Y. On Some Formulas for Kaprekar Constants. Symmetry 2019, 11, 885.
Yamagami A, Matsui Y. On Some Formulas for Kaprekar Constants. Symmetry. 2019; 11(7):885.
Yamagami, Atsushi; Matsui, Yūki. 2019. "On Some Formulas for Kaprekar Constants." Symmetry 11, no. 7: 885.
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