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Asymptotic behavior of the irrational factor

Acta Mathematica Hungarica
Volume/Issue:
Volume 121: Issue 3
DOI:
https://doi.org/10.1007/s10474-008-7212-9
Authors: E. Alkan, A. Ledoan, and A. Zaharescu

Abstract  

We study the irrational factor function I(n) introduced by Atanassov and defined by

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$I(n) = \prod\nolimits_{\nu = 1}^k {p_\nu ^{1/\alpha _\nu } }$$ \end{document}
, where
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$n = \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu } }$$ \end{document}
is the prime factorization of n. We show that the sequence {G(n)/n}n≧1, where G(n) = Πν=1 n I(ν)1/n, is convergent; this answers a question of Panaitopol. We also establish asymptotic formulas for averages of the function I(n).

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