Editing Multiply perfect number (section)

===Bi-unitary multiply perfect numbers===

A positive integer ''n'' is called a '''bi-unitary multi''' {{nowrap|''k''-'''perfect'''}} '''number'''  if σ<sup>**</sup>(''n'') = ''kn'' where σ<sup>**</sup>(''n'') is the sum of its [[bi-unitary divisor]]s. This concept is due to Peter Hagis (1987). A '''bi-unitary multiply perfect number''' is simply a bi-unitary multi {{nowrap|''k''-perfect}} number for some positive integer ''k''. Equivalently, bi-unitary multiply perfect numbers are those ''n'' for which ''n'' divides σ<sup>**</sup>(''n''). A bi-unitary multi {{nowrap|2-perfect}} number is naturally called a '''bi-unitary perfect number''', and a bi-unitary multi {{nowrap|3-perfect}} number is called a '''bi-unitary triperfect number'''.

A divisor ''d'' of a positive integer ''n'' is called a '''bi-unitary divisor''' of ''n'' if the greatest common unitary divisor (gcud) of ''d'' and ''n''/''d'' equals 1. This concept  is due to D. Surynarayana (1972). The sum of the (positive) bi-unitary divisors of ''n'' is denoted by σ<sup>**</sup>(''n''). 

Peter Hagis (1987) proved that there are no odd bi-unitary multiperfect numbers other than 1. Haukkanen and Sitaramaiah (2020) found all bi-unitary triperfect numbers of the form 2<sup>''a''</sup>''u'' where 1 ≤ ''a'' ≤ 6 and ''u'' is odd,<ref name="HS2020a">{{harvnb|Haukkanen|Sitaramaiah|2020a}}</ref><ref name="HS2020b">{{harvnb|Haukkanen|Sitaramaiah|2020b}}</ref><ref name="HS2020c">{{harvnb|Haukkanen|Sitaramaiah|2020c}}</ref> and partially the case where ''a'' = 7.<ref name="HS2020d">{{harvnb|Haukkanen|Sitaramaiah|2020d}}</ref>
<ref name="HS2021a">{{harvnb|Haukkanen|Sitaramaiah|2021a}}</ref>
Further, they fixed completely the case ''a'' = 8.<ref name="HS2021b">{{harvnb|Haukkanen|Sitaramaiah|2021b}}</ref>
Tomohiro Yamada (Determining all biunitary triperfect numbers of a certain form, arXiv:2406.19331 [math.NT], 2024) proved that 2160 = 3<sup>3</sup> 80 is the only biunitary triperfect number of the form 3<sup>''a''</sup>''u'' where 3 ≤ ''a''  and ''u'' is not divisible by 3.

The first few bi-unitary multiply perfect numbers are:
:1, 6, 60, 90, 120, 672, 2160, 10080, 22848, 30240 {{OEIS|A189000}}