High degree $b$-Niven numbers

Abstract

Let $b$ be a numeration base. A $b$-Niven number is one that is divisible by the sum of its base $b$ digits. We introduce high degree $b$-Niven numbers. These are $b$-Niven numbers that have a power greater than $1$ that is $b$-Niven number. Our main result shows that for each degree there exists an infinite set of bases $b$ for which $b$-Niven numbers of that degree exist. The high degree $b$-Niven numbers are given by explicit formulas and have all digits different from zero.