Reverse divisors are Ball numbers

Abstract

This study investigates the relationships between Ball's magic numbers and reverse divisors. More specifically, we extend the set of numbers that are simultaneously reverse divisors and Ball's magic numbers. Webster and Williams (2012) classify reverse divisors into three types: basic of first and second types, and non-basic ones.
The first type basics are numbers that begin with the digits 10 and end with 89, with all intermediate digits being 9, for example, 1099989. The second type basic is double to a first type basic, such as 2199978. Non-basic reverse divisors are obtained by concatenating magic reverse divisors. Costa and Santos (2024) proved that a first type basic reverse divisor is a Ball's magic number.
This work demonstrated that a non-basic reverse divisor, obtained by concatenating first type basic reverse divisors, is also a Ball's magic number.