SIMULTANEOUS NIVEN NUMBERS IN ARITHMETIC PROGRESSIONS FOR POWER-RELATED BASES

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 $b\ge 2$ . 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 $b^k$ -Niven (indeed, we can obtain simultaneous $b^\ell $ -Niven-ness for $\ell =1,\ldots , k$ ).