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

Citations:

Zbl 1023.11003; Zbl 1155.11345
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References:

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