MathDB

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

Problem Statement