On the counting function for the generalized Niven numbers
Ryan Daileda1 ; Jessica Jou2 ; Robert Lemke-Oliver3 ; Elizabeth Rossolimo4 ; Enrique Treviño5
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.

Étant donnés q≥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≤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≥2 and a completely q-additive arithmetic function f taking integer values, we deduce an asymptotic expression for the counting function

Nf(x)=#0≤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.

DOI : 10.5802/jtnb.685
Classification : 11A25, 11A63, 11K65
Ryan Daileda 1 ; Jessica Jou 2 ; Robert Lemke-Oliver 3 ; Elizabeth Rossolimo 4 ; Enrique Treviño 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 
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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, Tome 21 (2009) no. 3, pp. 503-515. doi : 10.5802/jtnb.685. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.685/

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