Even-length palindromes are weak additive Ramanujan-Hardy numbers, squares of palindromes with at least two digits are weak multiplicative Ramanujan-Hardy numbers, and fixed extra terms admit only finitely many such numbers.
High degree $b$-Niven numbers
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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.
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About Some Relatives of Palindromes
Even-length palindromes are weak additive Ramanujan-Hardy numbers, squares of palindromes with at least two digits are weak multiplicative Ramanujan-Hardy numbers, and fixed extra terms admit only finitely many such numbers.