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 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 is unknown whether there are any odd multiply perfect numbers other than 1. The first few multiply perfect numbers are:

1, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, ... (sequence A007691 in the OEIS).

Example

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.

Smallest known k -perfect numbers

The following table gives an overview of the smallest known k-perfect numbers for k ≤ 11 (sequence A007539 in the OEIS):

Properties

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.

Odd multiply perfect numbers

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:

• 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 o ( x ε ) {\displaystyle o(x^{\varepsilon })} for all ε > 0.

The number of k-perfect numbers n for n ≤ x is less than c x c ′ log ⁡ log ⁡ log ⁡ x / log ⁡ log ⁡ x {\displaystyle cx^{c'\log \log \log x/\log \log x}}, where c and c' are constants independent of k.

Under the assumption of the Riemann hypothesis, the following inequality is true for all k-perfect numbers n, where k > 3

log ⁡ log ⁡ n > k ⋅ e − γ {\displaystyle \log \log n>k\cdot e^{-\gamma }}

where γ {\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, where k > 2, satisfies the inequality

τ ( n ) > e k − γ . {\displaystyle \tau (n)>e^{k-\gamma }.}

The number of distinct prime factors ω(n) of n satisfies

ω ( n ) ≥ k 2 − 1 , if n is odd {\displaystyle \omega (n)\geq k^{2}-1,~~{\text{if }}n{\text{ is odd}}}

ω ( n ) ≥ k 2 / 4 , if n is even {\displaystyle \omega (n)\geq k^{2}/4,~~{\text{if }}n{\text{ is even}}}

If the distinct prime factors of n are p 1 , p 2 , … , p r {\displaystyle p_{1},p_{2},\ldots ,p_{r}}, then:

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

r ( k 2 3 r − 1 ) < ∑ i = 1 r 1 p i < r ( 1 − 8 / ( k π 2 ) r ) , if n is odd {\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}}}

Specific values of k

Perfect numbers

A number n with σ(n) = 2n is perfect.

Triperfect numbers

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.

Variations

Unitary multiply perfect numbers

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.

The first few unitary multiply perfect numbers are:

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

Bi-unitary multiply perfect numbers

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. 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.

In 2020, Haukkanen and Sitaramaiah studied bi-unitary triperfect numbers of the form 2au 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 3au 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)

Sources

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See also

• Hemiperfect number

External links

• on YouTube