Chebyshev’s inequality and natural density. (English) Zbl 0694.10004

For a function f: \({\mathbb{N}}_ 0\to {\mathbb{N}}\) let \(S_ f\) be the set of nonnegative integers n such that f(n) divides n and let \(S(x)=\#\{n\in S_ f, n<x\}\). Generalizing a method of an earlier paper [College Math. J. 15, 309-312 (1984)] the authors give sufficient conditions for the set \(S_ f\) to have natural density zero. The proof uses Chebyshev’s inequality. Besides the Niven numbers (i.e. f(n) denotes the digital sum of n) which were also studied in the above mentioned paper, other examples are treated.
Reviewer: Th.Maxsein


11A25 Arithmetic functions; related numbers; inversion formulas
11B83 Special sequences and polynomials
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