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

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.