Multiply perfect number

In mathematics, a multiply perfect number is a generalization of a perfect number.
For a given natural number k, a number n is called if the sum of all positive divisors of 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:

Example

The sum of the divisors of 120 is
which is 3 × 120. Therefore 120 is a number.

Smallest known ''k''-perfect numbers

The following table gives an overview of the smallest known numbers for k ≤ 11 :

k	Smallest k-perfect number	Factors	Found by
1	1		ancient
2	6	2 × 3	ancient
3	120	2³ × 3 × 5	ancient
4	30240	2⁵ × 3³ × 5 × 7	René Descartes, circa 1638
5	14182439040	2⁷ × 3⁴ × 5 × 7 × 11² × 17 × 19	René Descartes, circa 1638
6	154345556085770649600	2¹⁵ × 3⁵ × 5² × 7² × 11 × 13 × 17 × 19 × 31 × 43 × 257	Robert Daniel Carmichael, 1907
7	141310897947438348259849...523264343544818565120000	2³² × 3¹¹ × 5⁴ × 7⁵ × 11² × 13² × 17 × 19³ × 23 × 31 × 37 × 43 × 61 × 71 × 73 × 89 × 181 × 2141 × 599479	TE Mason, 1911
8	826809968707776137289924...057256213348352000000000	2⁶² × 3¹⁵ × 5⁹ × 7⁷ × 11³ × 13³ × 17² × 19 × 23 × 29 ×... × 487 × 521² × 601 × 1201 × 1279 × 2557 × 3169 × 5113 × 92737 × 649657	Stephen F. Gretton, 1990
9	561308081837371589999987...415685343739904000000000	2¹⁰⁴ × 3⁴³ × 5⁹ × 7¹² × 11⁶ × 13⁴ × 17 × 19⁴ × 23² × 29 ×... × 17351 × 29191 × 30941 × 45319 × 106681 × 110563 × 122921 × 152041 × 570461 × 16148168401	Fred Helenius, 1995
10	448565429898310924320164...000000000000000000000000	2¹⁷⁵ × 3⁶⁹ × 5²⁹ × 7¹⁸ × 11¹⁹ × 13⁸ × 17⁹ × 19⁷ × 23⁹ × 29³ ×... × 583367 × 1609669 × 3500201 × 119782433 × 212601841 × 2664097031 × 2931542417 × 43872038849 × 374857981681 × 4534166740403	George Woltman, 2013
11	312633142338546946283331...000000000000000000000000	2⁴¹³ × 3¹⁴⁵ × 5⁷³ × 7⁴⁹ × 11²⁷ × 13²² × 17¹¹ × 19¹³ × 23¹⁰ × 29⁹ ×... × 31280679788951 × 42166482463639 × 45920153384867 × 9460375336977361 × 18977800907065531 × 79787519018560501 × 455467221769572743 × 2519545342349331183143 × 38488154120055537150068589763279 × 6113142872404227834840443898241613032969	George Woltman, 2022

Properties

It can be proven that:

For a given prime number p, if n is and p does not divide n, then pn is. This implies that an integer n is a number divisible by 2 but not by 4, if and only if n/2 is an odd perfect number, of which none are known.
If 3n is and 3 does not divide n, then n is.
Odd multiply perfect numbers

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:

The largest prime factor is ≥ 100129
The second largest prime factor is ≥ 1009
The third largest prime factor is ≥ 101

If an odd triperfect number exists, it must be greater than 10¹²⁸.
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.

Bounds

In little-o notation, the number of multiply perfect numbers less than x is for all ε > 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 τ of a number n satisfies the inequality
The number of distinct prime factors ω of n satisfies
If the distinct prime factors of n are, then:

Specific values of ''k''

Perfect numbers

A number n with σ = 2n is perfect.

Triperfect numbers

A number n with σ = 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 then 2m would be, since σ = σσ = 3×2m. An odd triperfect number must be a square number exceeding 10⁷⁰ and have at least 12 distinct prime factors, the largest exceeding 10⁵.

Variations

Unitary multiply perfect numbers

A similar extension can be made for unitary perfect numbers. A positive integer n is called a unitary multi 'number if σ^* = kn where σ^* 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 10¹⁰² and must have at least 45 odd prime factors.
The first few unitary multiply perfect numbers are:

Bi-unitary multiply perfect numbers

A positive integer n is called a bi-unitary multi 'number if σ^** = kn where σ^** 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 2a''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 = 3³. This means that Yamada found all biunitary triperfect numbers of the form 3^au with 3 ≤ a and u not divisible by 3.
The first few bi-unitary multiply perfect numbers are: