v-Palindromes: An Analogy to the Palindromes

Around the year 2007, one of the authors, Tsai, accidentally discovered a property of the number $198$ he saw on the license plate of a car. Namely, if we take $198$ and its reversal $891$, which have prime factorizations 198 = 2· 3²· 11 and 891 = 3⁴· 11 respectively, and sum the numbers appearing in each factorization getting $2+3+2+11 = 18$ and $3+4+11 = 18$, both sums are $18$. Such numbers were 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. We also exhibit infinite families of v-palindromes in bases $p+1$ and p²+1, for each odd prime p. Finally, we collect some conjectures and problems involving v-palindromes.