v-palindromes: an analogy to the palindromes

Abstract

In the year 2007, the author discovered an intriguing property of the number $198$ he saw on the license plate of a car. Namely, if we take $198$ and its reversal $891$, prime factorize each number, and sum the numbers appearing in each factorization, both sums are $18$. Such numbers are formally introduced in a short published note in 2018. These numbers are later named $v$-palindromes because they can be viewed as an analogy to the usual palindromes. In this article, we introduce the concept of a $v$-palindrome in base $b$, and prove their existence for infinitely many bases. Finally, we collect some conjectures on $v$-palindromes.