Exponential unitary divisors

Abstract. We say that \(d\) is an exponential unitary divisor of \(n=p_1^{a_1}\cdots p_r^{a_r}>1\) if \(d=p_1^{b_1}\cdots p_r^{b_r}\), where \(b_i\) is a unitary divisor of \(a_i\), i.e., \(b_i\mid a_i\) and \((b_i,a_i/b_i)=1\) for every \(i\in \{1,2,\ldots,r\}\). We survey properties of related arithmetical functions and introduce the notion of exponential unitary perfect numbers.

Key words and phrases. Unitary divisor, exponential divisor, number of divisors, sum of divisors, Euler's function, perfect number.