About Some Relatives of Palindromes

TL;DR

This paper introduces two new classes of integers related to palindromes, exploring their properties and relationships to existing classes like b-ARH and b-MRH numbers, expanding the understanding of palindrome-related number classes.

Contribution

It defines two novel classes of integers based on digit sum and reversal operations, extending the framework of palindrome-related number classes and establishing their properties.

Findings

First class includes certain non-palindrome numbers with specific digit sum properties.

Second class contains squares of multi-digit palindromes.

These classes are larger than previously known b-ARH and b-MRH classes.

Abstract

We introduce two new classes of integers. The first class consists of numbers $N$ for which there exists at least one nonnegative integer $A$, such that the sum of $A$ and the sum of digits of $N$, added to the reversal of the sum, gives $N$. The second class consists of numbers $N$ for which there exists at least one nonnegative integer $A$, such that the sum of $A$ and the sum of the digits of $N$, multiplied by the reversal of the sum, gives $N$. All palindromes that either have an even number of digits or an odd number of digits and the middle digit even belong to the first class, and all squares of palindromes with at least two digits belong to the second class. These classes contain and are strictly larger than the classes of $b$-ARH numbers, respectively $b$-MRH numbers introduced in Ni\c tic\u a \cite{N1}.