Bi-unitary multiperfect numbers, V

Abstract

A divisor $d$ of a positive integer $n$ is called a unitary divisor if $\gcd(d, n/d)=1;$ and $d$ is called a bi-unitary divisor of $n$ if the greatest common unitary divisor of $d$ and $n/d$ is unity. The concept of a bi-unitary divisor is due to D. Surynarayana (1972). Let $\sig^{**}(n)$ denote the sum of the bi-unitary divisors of $n$. A positive integer $n$ is called a bi-unitary multiperfect number if $\sig^{**}(n)=kn$ for some $k\geq 3$. For $k=3$ we obtain the bi-unitary triperfect numbers. Peter Hagis (1987) proved that there are no odd bi-unitary multiperfect numbers. The present paper is part V in a series of papers on even bi-unitary multiperfect numbers. In parts I, II and III we determined all bi-unitary triperfect numbers of the form $n=2^{a}u$, where $1\leq a \leq 6$ and $u$ is odd. In parts IV(a-b) we solved partly the case $a=7$. In this paper we fix the case $a=8$. In fact, we show that $n=57657600=2^{8}.3^{2}.5^{2}.7.11.13$ is the only bi-unitary triperfect number of the present type.