Journal of Integer Sequences, Vol. 25 (2022), Article 22.6.7

Fixed Points and Cycles of the Kaprekar Transformation: 1. Odd Bases

Anthony Kay
72 Tiverton Road
Leicestershire LE11 2RY
United Kingdom

Katrina Downes-Ward


We extend Yamagami and Matsui's theory of the Kaprekar transformation in base 3 to higher odd bases. The structure of primitive proper, primitive non-proper and general cycles appears in all odd bases, but many other features differ according to whether the base is congruent to 1 or 3 modulo 4. In the latter case, cycles are derived from subgroups and cosets in multiplicative groups modulo odd numbers. We examine cycles and fixed points in bases 5 and 7 in some detail, and make some broad observations relating to higher bases. There also exist many cycles outside the primitive/general structure.

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(Concerned with sequences A000010 A003558 A165002 A165003 A165041 A165042 A165080 A165081 A165119 A165120.)

Received April 21 2022; revised versions received July 17 2022; July 18 2022; July 21 2022. Published in Journal of Integer Sequences, July 22 2022.

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