On the counting function for the generalized Niven numbers
For an integer \(q\geq 2\), a positive integer \(n\) is said to be a \(q\)-Niven (or \(q\)-Harshad) number if \(n\) is divisible by the sum of its digits in base \(q\), and more generally a positive integer \(n\) is said to be an \(f\)-Niven number if \(f(n)\mid n\), where \(f\) is a nonzero completely \(q\)-additive arithmetic function with integer values. The purpose of this paper is to prove an asymptotic formula for the number of \(f\)-Niven numbers not exceeding \(x\) under some mild conditions on \(f\). If \(f\) is the sum of digits function to the base \(q\), the result gives the corresponding asymptotic formulas by \textit{J.-M. De~Koninck, N. Doyon} and \textit{I. Kátai} [Acta Arith. 106, No.~3, 265--275 (2003; Zbl 1023.11003)] and \textit{C. Mauduit, C. Pomerance} and \textit{A. Sárközy} [Ramanujan J. 9, No.~1--2, 45--62 (2005; Zbl 1155.11345)]. Generalization to \(f\)-Niven numbers was suggested by J.-M. De~Koninck, N. Doyon and I. Kátai. The significant modifications needed in this general case are also pointed out in the paper under review.
completely \(q\)-additive arithmetic function
scientific article; zbMATH DE number 5774794