Top Qs
Timeline
Chat
Perspective

Multiply perfect number

Number whose divisors add to a multiple of that number From Wikipedia, the free encyclopedia

Multiply perfect number
Remove ads

In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number.

Thumb
Demonstration, with Cuisenaire rods, of the 2-perfection of the number 6

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.[1]

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).
Remove ads

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

Summarize
Perspective

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

More information k, Smallest k-perfect number ...
Remove ads

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

Unsolved problem in mathematics
Are there any 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:[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.

Remove ads

Bounds

Summarize
Perspective

In little-o notation, the number of multiply perfect numbers less than x is for all ε > 0.[2]

The number of k-perfect numbers n for nx is less than , 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 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]

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

If the distinct prime factors of n are , then:[6]

Remove ads

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.[7]

Remove ads

Variations

Summarize
Perspective

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.[8]

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.[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 2au 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]

The first few bi-unitary multiply perfect numbers are:

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

References

See also

Loading related searches...

Wikiwand - on

Seamless Wikipedia browsing. On steroids.

Remove ads