Problem

Source: Iran TST 2023 ; Exam 1 Problem 3

Tags: combinatorics



Arman, starting from a number, calculates the sum of the cubes of the digits of that number, and again calculates the sum of the cubes of the digits of the resulting number and continues the same process. Arman calls a number $Good$ if it reaches $1$ after performing a number of steps. Prove that there is an arithmetic progression of length $1402$ of good numbers. Proposed by Navid Safaei