Multiply perfect number

The sum of the divisors of 120 is

1 + 2 + 3 + 4 + 5 + 6 + 8 + 10 + 12 + 15 + 20 + 24 + 30 + 40 + 60 + 120 = 360

which is 3 × 120. Therefore 120 is a 3-perfect number.

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:

• For a given prime number p, if n is p-perfect and p does not divide n, then pn is (p + 1)-perfect. This implies that an integer n is a 3-perfect 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 4k-perfect and 3 does not divide n, then n is 3k-perfect.

It is unknown whether there are any odd multiply perfect numbers other than 1. However if an odd k-perfect number n exists where k > 2, then it must satisfy the following conditions:^[2]

• 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.^[3]

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.^[4] This is closely related to the concept of Descartes numbers.

In little-o notation, the number of multiply perfect numbers less than x is ${\displaystyle o(x^{\varepsilon })}$ for all ε > 0.^[2]

The number of k-perfect numbers n for n ≤ x is less than ${\displaystyle cx^{c'\log \log \log x/\log \log x}}$, 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

${\displaystyle \log \log n>k\cdot e^{-\gamma }}$

where ${\displaystyle \gamma }$ is Euler's gamma constant. This can be proven using Robin's theorem.

The number of divisors τ(n) of a k-perfect number n satisfies the inequality^[5]

${\displaystyle \tau (n)>e^{k-\gamma }.}$

The number of distinct prime factors ω(n) of n satisfies^[6]

${\displaystyle \omega (n)\geq k^{2}-1.}$

If the distinct prime factors of n are ${\displaystyle p_{1},p_{2},\ldots ,p_{r}}$, then:^[6]

${\displaystyle r\left({\sqrt[{r}]{3/2}}-1\right)<\sum _{i=1}^{r}{\frac {1}{p_{i}}}<r\left(1-{\sqrt[{r}]{6/k^{2}}}\right),~~{\text{if }}n{\text{ is even}}}$

${\displaystyle r\left({\sqrt[{3r}]{k^{2}}}-1\right)<\sum _{i=1}^{r}{\frac {1}{p_{i}}}<r\left(1-{\sqrt[{r}]{8/(k\pi ^{2})}}\right),~~{\text{if }}n{\text{ is odd}}}$

A number n with σ(n) = 2n is perfect.

A number n with σ(n) = 3n is triperfect. There are only six known triperfect numbers and these are believed to comprise all such numbers:

120, 672, 523776, 459818240, 1476304896, 51001180160 (sequence A005820 in the OEIS)

If there exists an odd perfect number m (a famous open problem) then 2m would be 3-perfect, since σ(2m) = σ(2)σ(m) = 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.^[7]

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 a unitary multi k-perfect number for some positive integer k. A unitary multi 2-perfect number is also called a unitary perfect number.

In the case k > 2, no example of a unitary multi k-perfect 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.^[8]

The first few unitary multiply perfect numbers are:

1, 6, 60, 90, 87360 (sequence A327158 in the OEIS)

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 a bi-unitary multi k-perfect number for some positive integer k.^[9] A bi-unitary multi 2-perfect number is also called a bi-unitary perfect number, and a bi-unitary multi 3-perfect 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.^[9]

In 2020, Haukkanen and Sitaramaiah studied bi-unitary triperfect numbers of the form 2^au where u is odd. They completely resolved the cases 1 ≤ a ≤ 6 and a = 8, and partially resolved the case a = 7.^{[10][11][12][13][14][15]}

In 2024, Tomohiro Yamada proved that 2160 is the only bi-unitary triperfect number divisible by 27 = 3³.^[16] 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:

1, 6, 60, 90, 120, 672, 2160, 10080, 22848, 30240 (sequence A189000 in the OEIS)

• Hemiperfect number