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 (or perfect) if the sum of all positive divisors of n (the divisor function, ÃÂ(n)) is equal to kn; a number is thus perfect if and only if it is . A number that is for a certain k is called a multiply perfect number. As of 2014, numbers are known for each value of k up to 11.
It is unknown whether there are any odd multiply perfect numbers other than 1. The first few multiply perfect numbers are:
The sum of the divisors of 120 is
which is 3 ÃÂ 120. Therefore 120 is a number.
The following table gives an overview of the smallest known numbers for k ⤠11 :
It can be proven that:
It is unknown whether there are any odd multiply perfect numbers other than 1. However if an odd number n exists where k > 2, then it must satisfy the following conditions:
If an odd triperfect number exists, it must be greater than 10<sup>128</sup>.
Tóth found several numbers that would be odd multiperfect, if one of their factors was a square. An example is 8999757, which would be an odd multiperfect number, if only one of its prime factors, 61, was a square. This is closely related to the concept of Descartes numbers.
In little-o notation, the number of multiply perfect numbers less than x is for all õ > 0.
The number of k-perfect numbers n for n ⤠x is less than , where c and c are constants independent of k.
Under the assumption of the Riemann hypothesis, the following inequality is true for all numbers n, where k > 3
where is Euler's gamma constant. This can be proven using Robin's theorem.
The number of divisors ÃÂ(n) of a number n, where k > 2, satisfies the inequality
The number of distinct prime factors ÃÂ(n) of n satisfies
If the distinct prime factors of n are , then:
A number n with ÃÂ(n) = 2n is perfect.
A number n with ÃÂ(n) = 3n is triperfect. There are only six known triperfect numbers and these are believed to comprise all such numbers:
If there exists an odd perfect number m (a famous open problem) then 2m would be , since ÃÂ(2m) = ÃÂ(2)ÃÂ(m) = 3ÃÂ2m. An odd triperfect number must be a square number exceeding 10<sup>70</sup> and have at least 12 distinct prime factors, the largest exceeding 10<sup>5</sup>.
A similar extension can be made for unitary perfect numbers. A positive integer n is called a unitary multi number if ÃÂ<sup>*</sup>(n) = kn where ÃÂ<sup>*</sup>(n) is the sum of its unitary divisors. A unitary multiply perfect number is a unitary multi number for some positive integer k. A unitary multi number is also called a unitary perfect number.
In the case k > 2, no example of a unitary multi number is yet known. It is known that if such a number exists, it must be even and greater than 10<sup>102</sup> and must have at least 45 odd prime factors.
The first few unitary multiply perfect numbers are:
A positive integer n is called a bi-unitary multi number if ÃÂ<sup>**</sup>(n) = kn where ÃÂ<sup>**</sup>(n) is the sum of its bi-unitary divisors. A bi-unitary multiply perfect number is a bi-unitary multi number for some positive integer k. A bi-unitary multi number is also called a bi-unitary perfect number, and a bi-unitary multi number is called a bi-unitary triperfect number.
In 1987, Peter Hagis proved that there are no odd bi-unitary multiperfect numbers other than 1.
In 2020, Haukkanen and Sitaramaiah studied bi-unitary triperfect numbers of the form 2<sup>a</sup>u where u is odd. They completely resolved the cases 1 ⤠a ⤠6 and a = 8, and partially resolved the case a = 7.
In 2024, Tomohiro Yamada proved that 2160 is the only bi-unitary triperfect number divisible by 27 = 3<sup>3</sup>. This means that Yamada found all biunitary triperfect numbers of the form 3<sup>a</sup>u with 3 ⤠a and u not divisible by 3.
The first few bi-unitary multiply perfect numbers are: