## On the counting function for the Niven numbers.(English)Zbl 1023.11003

Given an integer $$q\geq 2$$, a positive integer $$n$$ is said to be a $$q$$-Niven number if it is divisible by the sum of its digits in base $$q$$. Let $$N_q(x)$$ denote the number of $$q$$-Niven numbers not exceeding $$x$$. The main purpose of this paper is to prove that $$N_q(x)=(\eta_q + o(1)){x\over \log x}$$ as $$x\to\infty$$, where $$\eta_q={2\log q\over (q-1)^2}\sum_{j=1}^{q-1}(j, q-1)$$. For $$q=10$$, this shows that the conjecture posed by J.-M. De Koninck and N. Doyon [Fibonacci Q. 41, No. 5, 431–440 (2003; Zbl 1057.11005)] holds.

### MSC:

 11A25 Arithmetic functions; related numbers; inversion formulas 11A63 Radix representation; digital problems 11K65 Arithmetic functions in probabilistic number theory 11N37 Asymptotic results on arithmetic functions

Zbl 1057.11005
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