Multiply perfect number

[1] It is unknown whether there are any odd multiply perfect numbers other than 1.

The following table gives an overview of the smallest known k-perfect numbers for k ≤ 11 (sequence A007539 in the OEIS): It can be proven that: It is unknown whether there are any odd multiply perfect numbers other than 1.

, where c and c' are constants independent of k.[2] Under the assumption of the Riemann hypothesis, the following inequality is true for all k-perfect numbers n, where k > 3 where

[5] A similar extension can be made for unitary perfect numbers.

A positive integer n is called a unitary multi k-perfect number if σ*(n) = kn where σ*(n) is the sum of its unitary divisors.

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).

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.

The first few unitary multiply perfect numbers are: A positive integer n is called a bi-unitary multi k-perfect number if σ**(n) = kn where σ**(n) is the sum of its bi-unitary divisors.

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).

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 other than 1.

Haukkanen and Sitaramaiah (2020) found all bi-unitary triperfect numbers of the form 2au where 1 ≤ a ≤ 6 and u is odd,[6][7][8] and partially the case where a = 7.