How often is the number of divisors of n a divisor of n ?
type: Item
instance of: scholarly article
title: How often is the number of divisors of n a divisor of n ?
zbMATH Open document ID: 0568.10021
DOI: 10.1016/0022-314X(85)90012-5
published in: Journal of Number Theory
publication date: 1985-01-01T00:00:00Z
review text: Let d(n) denote the number of positive integers dividing the positive integer n. The author establishes (by ingenious arguments) the following results: Theorem 1. \[ \#\{n\leq x:\quad d(n) | n\}=(x/\sqrt{\log x})(\log \log x)^{-1+o(1)}. \] Theorem 2. Let \(\{\xi_ k\}^{\infty}_{k=0}\) be defined recursively by \(\xi_ 0=0\), \(\xi_ k=\xi_{k-1}+2^{\xi_{k-1}}\) for \(k\geq 1\). Then there is a constant C such that \[ \#\{n\leq x:\quad d(n) | n\}\quad \leq \frac{C x (\log \log \log \log x)^{1/2} \xi (x)}{(\log x)^{1/2} \log \log x} \] for \(x\geq e^{16}\), where \(\xi (x)=\#\{k\geq 0\), \(\xi_ k\leq x\}\).
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 3906585
zbMATH Keywords: number of divisors
zbMATH Keywords: cardinality
zbMATH Keywords: upper bound
author: Claudia A. Spiro
MaRDI profile type: Publication
Wikidata QID: Q60018622
cites work: Q3934484
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description: scientific article; zbMATH DE number 3906585