, Volume 121, Issue 3, pp 293-305
Date: 18 Sep 2008

Asymptotic behavior of the irrational factor

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Abstract

We study the irrational factor function I(n) introduced by Atanassov and defined by \( I(n) = \prod\nolimits_{\nu = 1}^k {p_\nu ^{1/\alpha _\nu } } \) , where \( n = \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu } } \) is the prime factorization of n. We show that the sequence {G(n)/n} n≧1, where G(n) = Π ν=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).

Research of the first author is supported in part by TUBITAK Career Award and Distinguished Young Scholar Award, TUBA-GEBIP of Turkish Academy of Sciences.
Research of the third author is supported in part by NSF grant number DMS-0456615.