On the counting function for the generalized Niven numbers

Daileda, Ryan; Jou, Jessica; Lemke-Oliver, Robert; Rossolimo, Elizabeth; Treviño, Enrique

doi:10.5802/jtnb.685

Daileda, Ryan; Jou, Jessica; Lemke-Oliver, Robert; Rossolimo, Elizabeth; Treviño, Enrique

Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, p. 503-515

Abstract
Résumé

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.

Étant donnés $q \geq 2$ un entier naturel et $f$ une fonction complètement q-additive à valeurs dans l’ensemble des nombres entiers relatifs, on calcule une expression asymptotique de la fonction $N_{f}$ qui à $x$ associe la cardinalité de l’ensemble

{0 \leq n < x | f (n) ∣ n}

quand les valeurs de $f$ sont soumises à une petite restriction. Dans le cas où $f = s_{q}$ , la somme des chiffres d’un nombre en base $q$ , les valeurs de la function $N_{f}$ comptent les nombres q-Harshad. Donc, notre résultat généralise la formule asymptotique dans ce cas.

MR 2605530 | Zbl 1205.11105

DOI : https://doi.org/10.5802/jtnb.685
Classification: 11A25, 11A63, 11K65

@article{JTNB_2009__21_3_503_0,
     author = {Daileda, Ryan and Jou, Jessica and Lemke-Oliver, Robert and Rossolimo, Elizabeth and Trevi\~no, Enrique},
     title = {On the counting function for the generalized Niven numbers},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {3},
     year = {2009},
     pages = {503-515},
     doi = {10.5802/jtnb.685},
     mrnumber = {2605530},
     zbl = {1205.11105},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2009__21_3_503_0}
}

Daileda, Ryan; Jou, Jessica; Lemke-Oliver, Robert; Rossolimo, Elizabeth; Treviño, Enrique. On the counting function for the generalized Niven numbers. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, pp. 503-515. doi : 10.5802/jtnb.685. http://www.numdam.org/item/JTNB_2009__21_3_503_0/

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