Abstract 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).
Scott Duke Kominers (Tue,) studied this question.