Multiply perfect number

thumb|Demonstration, with [[Cuisenaire rods, of the of the number 6]]

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 (or 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 . 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.

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| 9 || 561308081837371589999987...415685343739904000000000 (287 digits) || 2¹⁰⁴ × 3⁴³ × 5⁹ × 7¹² × 11⁶ × 13⁴ × 17 × 19⁴ × 23² × 29 × ... × 17351 × 29191 × 30941 × 45319 × 106681 × 110563 × 122921 × 152041 × 570461 × 16148168401 (66 distinct prime factors)|| Fred Helenius, 1995

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 <math>o(x^\varepsilon)</math> for all ε > 0.

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

:<math>\tau(n) > e^{k - \gamma}.</math>

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

:<math>\omega(n) \ge k^2-1, ~~ \text{if }n\text{ is odd}</math>

:<math>\omega(n) \ge k^2/4, ~~ \text{if }n\text{ is even}</math>

If the distinct prime factors of n are <math>p_1, p_2, \ldots, p_r</math>, then:

:<math>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}</math>

:<math>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}</math>

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

If there exists an odd perfect number m (a famous open problem) then 2m would be , 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 number if σ^*(n) = kn where σ^*(n) 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:

:1, 6, 60, 90, 87360

Bi-unitary multiply perfect numbers

A positive integer n is called a bi-unitary multi 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 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 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:

:1, 6, 60, 90, 120, 672, 2160, 10080, 22848, 30240

References

Sources

See also

• Hemiperfect number

External links

• The Multiply Perfect Numbers page
• The Prime Glossary: Multiply perfect numbers