Multiply perfect number

In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number.
For a given natural number "k", a number "n" is called "k"-perfect (or "k"-fold perfect) if and only if the sum of all positive divisors of "n" (the divisor function, "σ"("n")) is equal to "kn"; a number is thus perfect if and only if it is 2-perfect. A number that is "k"-perfect for a certain "k" is called a multiply perfect number. As of 2014, "k"-perfect numbers are known for each value of "k" up to 11.
It can be proven that:
Example.
The divisors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120. Their sum is 360, which equals 
formula_1, so 120 is 3-perfect.
Smallest "k"-perfect numbers.
The following table gives an overview of the smallest "k"-perfect numbers for "k" ≤ 11 :
Specific values of "k".
Perfect numbers.
A number "n" with σ("n") = 2"n" is perfect.
Triperfect numbers.
A number "n" with σ("n") = 3"n" is triperfect. An odd triperfect number must exceed 1070 and have at least 12 distinct prime factors, the largest exceeding 105.
Variations.
Unitary multiply perfect numbers.
A positive integer "n" is called a unitary multi "k"-perfect number if σ*("n") = "kn". A unitary multiply perfect number is simply a unitary multi "k"-perfect number for some positive integer "k". Equivalently, unitary multiply perfect numbers are those "n" for which "n" divides σ*("n"). A unitary multi 2-perfect number is naturally called a unitary perfect number. In the case "k" > 2, no example of a unitary multi "k"-perfect number is known till now. It is known that if such a number exists, it must be even and greater than 10102 and must have more than forty four odd prime factors. This problem is probably very difficult to settle.
A divisor "d" of a positive integer "n" is called a unitary divisor if gcd("d", "n"/"d") = 1. The concept of unitary divisor was originally due to R. Vaidyanathaswamy (1931) who called such a divisor as block factor. The present terminology is due to E. Cohen (1960). The sum of the (positive) unitary divisors of "n" is denoted by σ*("n").
Bi-unitary multiply perfect numbers.
A positive integer "n" is called a bi-unitary multi "k"-perfect number if σ**("n") = "kn". This concept is due to Peter Hagis (1987). A bi-unitary multiply perfect number is simply a bi-unitary multi "k"-perfect number for some positive integer "k". Equivalently, bi-unitary multiply perfect numbers are those "n" for which "n" divides σ**("n"). A bi-unitary multi 2-perfect number is naturally called a bi-unitary perfect number, and a bi-unitary multi 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**("n"). 
Peter Hagis (1987) proved that there are no odd bi-unitary multiperfect numbers. Haukkanen and Sitaramaiah (2020) found certain even bi-unitary triperfect numbers.