# Asymptotic behavior of the irrational factor

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s10474-008-7212-9

- Cite this article as:
- Alkan, E., Ledoan, A.H. & Zaharescu, A. Acta Math Hung (2008) 121: 293. doi:10.1007/s10474-008-7212-9

## 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*).