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A SIEVE PROBLEM AND ITS APPLICATION
Published online by Cambridge University Press: 27 September 2016
Abstract
Let $\unicode[STIX]{x1D703}$ be an arithmetic function and let
${\mathcal{B}}$ be the set of positive integers
$n=p_{1}^{\unicode[STIX]{x1D6FC}_{1}}\cdots p_{k}^{\unicode[STIX]{x1D6FC}_{k}}$ which satisfy
$p_{j+1}\leqslant \unicode[STIX]{x1D703}(p_{1}^{\unicode[STIX]{x1D6FC}_{1}}\cdots p_{j}^{\unicode[STIX]{x1D6FC}_{j}})$ for
$0\leqslant j<k$. We show that
${\mathcal{B}}$ has a natural density, provide a criterion to determine whether this density is positive, and give various estimates for the counting function of
${\mathcal{B}}$. When
$\unicode[STIX]{x1D703}(n)/n$ is non-decreasing, the set
${\mathcal{B}}$ coincides with the set of integers
$n$ whose divisors
$1=d_{1}<d_{2}<\cdots <d_{\unicode[STIX]{x1D70F}(n)}=n$ satisfy
$d_{j+1}\leqslant \unicode[STIX]{x1D703}(d_{j})$ for
$1\leqslant j<\unicode[STIX]{x1D70F}(n)$.
MSC classification
- Type
- Research Article
- Information
- Copyright
- Copyright © University College London 2016
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