On primitive unitary abundant numbers (Q582314)

A positive divisor d of a positive integer n is said to be a unitary divisor of n if d and n/d are relatively prime. The sum of all unitary divisors of n is symbolized by \(\sigma^*(n)\) and n is said to be unitary r-abundant, unitary r-perfect or unitary r-deficient according as \(\sigma^*(n)>rn\), \(\sigma^*(n)=rn\) or \(\sigma^*(n)<rn\), respectively. (Here, r is a real number greater than or equal to 2.) A unitary r-abundant number n is said to be primitive if each of its proper unitary divisors is unitary r-deficient. This paper is devoted to the study of \(U_ r\), the set of all primitive unitary r-abundant numbers. Thus, it is shown that if n is an element of \(U_ r\) then \(\sigma^*(n)-rn<rn/q^ b\) where \(q^ b\) is the largest prime power divisor of n. Also, it is proved that if n is an element of \(U_ r\) then \(\sigma^*(n)/n\) approaches r as n approaches infinity. It is assumed here that the limit of \(\sigma^*(n)/n\) exists and that \(U_ r\) is infinite. The latter is not difficult to prove and the former assumption can be avoided by replacing ``limit'' by ``limit superior'' in the proof. The paper contains a number of misprints and minor errors. For example, in inequality (3.10) ``269'' should be replaced by ``268.5''.