M-FUNCTION NUMBERS: CYCLES AND OTHER EXPLORATIONS. PART 1

Авторы

• Smaiyl Makyshov Phillips Exeter Academy, PEA #1650, Exeter, NH 03833, USA

Ключевые слова:

M-Function, D.R.Kaprekar, self-numbers.

Аннотация

This paper establishes the cyclic properties of the M-Function, which we define as a function, [M(n)], that takes a positive integer, adds to it the sum of its digits and the number produced by reversing its digits, and then divides the entire sum by three. Our definition of the M-Function is influenced by D. R. Kaprekar’s work on a remarkable class of positive integers, called self- numbers, and his procedure, [K(n)], of adding to any positive integer the sum of its digits [1]. We analyze the distribution of numbers that make the defined M-Function behave like a cyclic function, and observe that many such “cycles” form arithmetic sequences. We examine the distribution of numbers that produce integer ratios between the outputs of Kaprekar’s and the M-Function functions, [K(n)/M(n)]. We also prove that the set of numbers with equal outputs to both Kaprekar’s and M-Function functions, [K(n)=M(n)], is infinite.

2020-02-04

Как цитировать

Makyshov, S. (2020). M-FUNCTION NUMBERS: CYCLES AND OTHER EXPLORATIONS. PART 1. Известия НАН РК. Серия физико-математическая, (1), 46–54. извлечено от https://journals.nauka-nanrk.kz/physics-mathematics/article/view/391

Статьи