Simultaneous Niven Numbers in Arithmetic Progressions for Power-Related Bases

Recently, Harrington et al. [‘Every arithmetic progression contains infinitely many b -Niven numbers’, Bull. Aust. Math. Soc. 109 (3) (2024), 409–413] proved that every arithmetic progression contains infinitely many base- b Niven numbers for any fixed b≥ 2 . We use a sparse repunit construction to treat a structured two-base version of the same problem, showing that every arithmetic progression with common difference relatively prime to b contains infinitely many integers that are simultaneously b -Niven and bᵏ -Niven (indeed, we can obtain simultaneous b^ -Niven-ness for =1,… , k ).