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On the counting function for the generalized Niven numbers. (English) Zbl 1205.11105

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 J.-M. De Koninck, N. Doyon and I. Kátai [Acta Arith. 106, No. 3, 265–275 (2003; Zbl 1023.11003)] and C. Mauduit, C. Pomerance and 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.

MSC:

11N37 Asymptotic results on arithmetic functions
11N64 Other results on the distribution of values or the characterization of arithmetic functions
11A63 Radix representation; digital problems
11N56 Rate of growth of arithmetic functions
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References:

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