On the exponent average of integer sequences

Abstract

Let ω(m) denote the number of distinct prime factors of the integerm, let Ω(m) be the number of prime factors ofm counted with multiplicities. The exponent averageA(m) is defined byA(m)=Ω(m)/ω(m) form>1, andA(1)=1. If (m _n) is a sequence of positive integers, we can study the asymptotic exponent average lim _n→∞ A(m_n) (if it exists) resp. lim sup and lim inf.

In this article, we consider exponent averages for general sequences, and particularly for sequences of binomial coefficients as well as the divisor function. One of the many results on binomial coefficients is that

$$\mathop {\lim }\limits_{n \to \infty } A\left( {\left( {\begin{array}{*{20}c} {2n} \\ n \\ \end{array} } \right)} \right) = 1,$$

which shows that these binomial coefficients are almost squarefree. For the divisor functiond(n), we prove for instance

$$\mathop {\lim \sup }\limits_{n \to \infty } \frac{{A(d(n))\log \log n}}{{\log n}} = 1.$$