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 ¹ ; Jou, Jessica ² ; Lemke-Oliver, Robert ³ ; Rossolimo, Elizabeth ⁴ ; Treviño, Enrique ⁵

¹ Mathematics Department Trinity University One Trinity Place San Antonio, TX 78212-7200, USA
² 678-2 Azumi, Ichinomiya-cho Shiso-shi, Hyogo 671-4131 Japan
³ Deparment of Mathematics University of Wisconsin Madison 480 Lincoln Dr Madison, WI 53706 USA
⁴ Department of Mathematics and Statistics Lederle Graduate Research Tower Box 34515 University of Massachusetts Amherst Amherst, MA 01003-9305, USA
⁵ Department of Mathematics 6188 Kemeny Hall Dartmouth College Hanover, NH 03755-3551, USA

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

Résumé
Abstract

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.

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.

MR Zbl

DOI : 10.5802/jtnb.685

Classification : 11A25, 11A63, 11K65

Affiliations des auteurs :

Daileda, Ryan ¹ ; Jou, Jessica ² ; Lemke-Oliver, Robert ³ ; Rossolimo, Elizabeth ⁴ ; Treviño, Enrique ⁵

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     title = {On the counting function for the generalized {Niven} numbers},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {503--515},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
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     zbl = {1205.11105},
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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, Tome 21 (2009) no. 3, pp. 503-515. doi : 10.5802/jtnb.685. http://archive.numdam.org/articles/10.5802/jtnb.685/

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