On Junction Numbers

Abstract

A base b junction number u has the property that there are at least two ways to write it as u = v+s(v), where s(v) is the sum of the digits in the expansion of the number v in base b. For the base 10 case, Kaprekar in the 1950’s and 1960’s studied the problem of finding K(n), the smallest u such that the equation u = v + s(v) has exactly n solutions. He gave the values K(2) = 101, K(3) = 1013 + 1, and conjectured that K(4) = 1024 + 102. In 1966 Narasinga Rao gave the upper bound 101111111111124+102 for K(5), as well as upper bounds for K(6), K(7), K(8), and K(16). We show that all these values are the correct values of K(n), and we give an algorithm for recursively determining the values of K(n) for any base b. The key to our approach is an apparently new recurrence for F (u), the number of solutions to u = v + s(v). We give tables of K(n) in bases b ≤ 10. Rather surprisingly, the solution to the base 2 problem is determined by the classical Thue-Morse sequence.

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Cite this paper

@inproceedings{Alekseyev2014OnJN, title={On Junction Numbers}, author={Max A. Alekseyev and Donovan Johnson and N. J. A. Sloane and Ramchandra Kaprekar}, year={2014} }