Multiply perfect number

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 :

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^\varepsilon) for all ε 0.

The number of k-perfect numbers n for n ≤ x is less than 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\cdot e^{-\gamma} where \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 :\tau(n) e^{k - \gamma}.

The number of distinct prime factors ω(n) of n satisfies :\omega(n) \ge k^2-1.

If the distinct prime factors of n are p_1, p_2, \ldots, p_r, then: :r \left(\sqrt[r]{3/2} - 1\right) :r \left(\sqrt[3r]{k^2} - 1\right)

Specific values of ''k''

Perfect numbers

Main article: Perfect number

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

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

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 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 3a**u 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

References

Sources

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References

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8. (1984). "Lower Bounds for Unitary Multiperfect Numbers". The Fibonacci Quarterly.
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15. {{harvnb. Haukkanen. Sitaramaiah. 2021b
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