On arithmetic properties of integers with missing digits. II: Prime factors
type: Item
instance of: scholarly article
title: On arithmetic properties of integers with missing digits. II: Prime factors
zbMATH Open document ID: 0945.11006
DOI: 10.1016/S0012-365X(98)00331-8
author: Christian Mauduit
author: András Sárközy
author: P. Erdős
published in: Discrete Mathematics
publication date: 2000-05-08T00:00:00Z
review text: For an integer \(g\geq 3\) let \(\mathcal D\) be a subset of \(\{0, 1, ..., g-1\}\) with \(0\in \mathcal D\), and let \(\mathcal W\) be the set of those integers whose development in base \(g\) contains only digits from \(\mathcal D\). The paper continues the study of the arithmetic properties of the set \(\mathcal W\), started in part I [J. Number Theory 70, 99-120 (1998; Zbl 0923.11024)]. Write \({\mathcal D } = \{d_1, ..., d_t\}\). Under the assumptions \((d_1, ..., d_t)=1\) and \(t>\sqrt g\), the authors prove a limiting law for the distribution of \(\omega (n)\) for \(n\in \mathcal D\), \(n\leq N\), and show that the maximal order of \(\omega \) is \( \gg ( \log N)/ \log\log N \). They list exciting open problems, like finding odd and even values of \(\omega \) in \(\mathcal D\), or finding almost primes in this set. As a partial result concerning the second problem, they give estimates for the largest prime factor of elements of \(\mathcal D\).
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 11N69
Mathematics Subject Classification ID: 11N25
zbMATH DE Number: 1334447
zbMATH Keywords: digits
zbMATH Keywords: number systems
zbMATH Keywords: uniform distribution
Wikidata QID: Q105956336
reviewed by: Imre Z. Ruzsa
MaRDI profile type: Publication
cites work: On sums of sequences of integers. II,III
cites work: On the uniform distribution modulo one of some subsequences of polynomial sequences
cites work: On the uniform distribution modulo one of subsequences of polynomial sequences. II
cites work: Graphes connexes représentation des entiers et équirepartition
cites work: Q3867876
cites work: The distribution of the number of prime divisors of sums \(a+b\)
cites work: On the sum \(\sum_{d\mid 2^n-1} d^{-1}\)
cites work: Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law
cites work: Sums of digits and almost primes
cites work: Méthodes de crible et fonctions sommes des chiffres
cites work: Sur les nombres qui ont des propriétés additives et multiplicatives données
cites work: Automates finis et ensembles normaux. (Finite automata and normal sets)
cites work: On the arithmetic structure of sets characterized by sum of digits properties
cites work: On the arithmetic structure of the integers whose sum of digits is fixed
cites work: The analytic principle of the large sieve
cites work: Q4291376
cites work: Q3699787
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description: scientific article; zbMATH DE number 1334447