On the counting function for the generalized Niven numbers
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, p. 503-515

Given an integer base q2 and a completely q-additive arithmetic function f taking integer values, we deduce an asymptotic expression for the counting function

Nf(x)=#0n<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 q2 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

{0n<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.

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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