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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

Citations:

Zbl 1057.11005
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