On the counting function for the generalized Niven numbers
type: Item
instance of: scholarly article
title: On the counting function for the generalized Niven numbers
zbMATH Open document ID: 1205.11105
author: Jessica Jou
author: Elizabeth Rossolimo
author: Enrique Treviño
author: Ryan C. Daileda
author: Robert J. Lemke Oliver
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2010-08-25T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10895
review text: For an integer \(q\geq 2\), a positive integer \(n\) is said to be a \(q\)-Niven (or \(q\)-Harshad) number if \(n\) is divisible by the sum of its digits in base \(q\), and more generally a positive integer \(n\) is said to be an \(f\)-Niven number if \(f(n)\mid n\), where \(f\) is a nonzero completely \(q\)-additive arithmetic function with integer values. The purpose of this paper is to prove an asymptotic formula for the number of \(f\)-Niven numbers not exceeding \(x\) under some mild conditions on \(f\). If \(f\) is the sum of digits function to the base \(q\), the result gives the corresponding asymptotic formulas by \textit{J.-M. De~Koninck, N. Doyon} and \textit{I. Kátai} [Acta Arith. 106, No.~3, 265--275 (2003; Zbl 1023.11003)] and \textit{C. Mauduit, C. Pomerance} and \textit{A. Sárközy} [Ramanujan J. 9, No.~1--2, 45--62 (2005; Zbl 1155.11345)]. Generalization to \(f\)-Niven numbers was suggested by J.-M. De~Koninck, N. Doyon and I. Kátai. The significant modifications needed in this general case are also pointed out in the paper under review.
reviewed by: Pentti Haukkanen
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11N64
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 11N56
zbMATH DE Number: 5774794
zbMATH Keywords: Niven number
zbMATH Keywords: Harshad number
zbMATH Keywords: sum of digits function
zbMATH Keywords: completely \(q\)-additive arithmetic function
zbMATH Keywords: asymptotic formula
MaRDI profile type: Publication
OpenAlex ID: W1963979652
cites work: On an asymptotic formula for the Niven numbers
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DOI: 10.5802/JTNB.685
description: scientific article; zbMATH DE number 5774794