Daileda, Jou, Lemke-Oliver: On the counting function for the generalized Niven numbers

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Ryan Daileda; Jessica Jou; Robert Lemke-Oliver; Elizabeth Rossolimo; Enrique Treviño
On the counting function for the generalized Niven numbers
Journal de théorie des nombres de Bordeaux, 21 no. 3 (2009), p. 503-515
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Class. Math.: 11A25, 11A63, 11K65

Résumé - Abstract

Given an integer base $q \geq 2$ and a completely $q$ -additive arithmetic function $f$ taking integer values, we deduce an asymptotic expression for the counting function

$N_{f} (x) = # \{0 \leq n < x | f (n) ∣ n\}$

under a mild restriction on the values of $f$ . When $f = s_{q}$ , the base $q$ sum of digits function, the integers counted by $N_{f}$ are the so-called base $q$ Niven numbers, and our result provides a generalization of the asymptotic known in that case.

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