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 Template:Nowrap (or Template:Nowrap 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 Template:Nowrap. A number that is Template:Nowrap for a certain k is called a multiply perfect number. As of 2014, Template:Nowrap numbers are known for each value of k up to 11.<ref name=fl/>
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, ... Template: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 Template:Nowrap number.
Smallest known k-perfect numbers
The following table gives an overview of the smallest known Template:Nowrap numbers for k ≤ 11 Template:OEIS:
| k | Smallest k-perfect number | Factors | Found by |
|---|---|---|---|
| 1 | 1 | ancient | |
| 2 | 6 | 2 × 3 | ancient |
| 3 | 120 | 23 × 3 × 5 | ancient |
| 4 | 30240 | 25 × 33 × 5 × 7 | René Descartes, circa 1638 |
| 5 | 14182439040 | 27 × 34 × 5 × 7 × 112 × 17 × 19 | René Descartes, circa 1638 |
| 6 | 154345556085770649600 (21 digits) | 215 × 35 × 52 × 72 × 11 × 13 × 17 × 19 × 31 × 43 × 257 | Robert Daniel Carmichael, 1907 |
| 7 | 141310897947438348259849...523264343544818565120000 (57 digits) | 232 × 311 × 54 × 75 × 112 × 132 × 17 × 193 × 23 × 31 × 37 × 43 × 61 × 71 × 73 × 89 × 181 × 2141 × 599479 | TE Mason, 1911 |
| 8 | 826809968707776137289924...057256213348352000000000 (133 digits) | 262 × 315 × 59 × 77 × 113 × 133 × 172 × 19 × 23 × 29 × ... × 487 × 5212 × 601 × 1201 × 1279 × 2557 × 3169 × 5113 × 92737 × 649657 (38 distinct prime factors) | Stephen F. Gretton, 1990<ref name=fl>Template:Cite web</ref> |
| 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<ref name=fl/> |
| 10 | 448565429898310924320164...000000000000000000000000 (639 digits) | 2175 × 369 × 529 × 718 × 1119 × 138 × 179 × 197 × 239 × 293 × ... × 583367 × 1609669 × 3500201 × 119782433 × 212601841 × 2664097031 × 2931542417 × 43872038849 × 374857981681 × 4534166740403 (115 distinct prime factors) | George Woltman, 2013<ref name=fl/> |
| 11 | 312633142338546946283331...000000000000000000000000 (1739 digits) | 2413 × 3145 × 573 × 749 × 1127 × 1322 × 1711 × 1913 × 2310 × 299 × ... × 31280679788951 × 42166482463639 × 45920153384867 × 9460375336977361 × 18977800907065531 × 79787519018560501 × 455467221769572743 × 2519545342349331183143 × 38488154120055537150068589763279 × 6113142872404227834840443898241613032969 (241 distinct prime factors) | George Woltman, 2022<ref name=fl/> |
Properties
It can be proven that:
- For a given prime number p, if n is Template:Nowrap and p does not divide n, then pn is Template:Nowrap. This implies that an integer n is a Template:Nowrap 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 Template:Nowrap and 3 does not divide n, then n is Template:Nowrap.
Odd multiply perfect numbers
Template:Unsolved It is unknown whether there are any odd multiply perfect numbers other than 1. However if an odd Template:Nowrap number n exists where k > 2, then it must satisfy the following conditions:<ref name="HBI105" />
- 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.<ref>Template:Cite web</ref>
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.<ref>Template:Harvtxt</ref> 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.<ref name="HBI105">Template:Harvnb</ref>
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.<ref name="HBI105" />
Under the assumption of the Riemann hypothesis, the following inequality is true for all Template:Nowrap numbers n, where k > 3
- <math>\log\log n > k\cdot e^{-\gamma}</math>
where <math>\gamma</math> is Euler's gamma constant. This can be proven using Robin's theorem.
The number of divisors τ(n) of a Template:Nowrap number n satisfies the inequality<ref>Template:Cite arXiv</ref>
- <math>\tau(n) > e^{k - \gamma}.</math>
The number of distinct prime factors ω(n) of n satisfies<ref name="HBI106">Template:Harvnb</ref>
- <math>\omega(n) \ge k^2-1.</math>
If the distinct prime factors of n are <math>p_1, p_2, \ldots, p_r</math>, then:<ref name="HBI106" />
- <math>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}</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
Template:Main 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 Template:OEIS
If there exists an odd perfect number m (a famous open problem) then 2m would be Template:Nowrap, 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.<ref>Template:Harvnb</ref>
Variations
Unitary multiply perfect numbers
A similar extension can be made for unitary perfect numbers. A positive integer n is called a unitary multi Template:Nowrap number if σ*(n) = kn where σ*(n) is the sum of its unitary divisors. A unitary multiply perfect number is a unitary multi Template:Nowrap number for some positive integer k. A unitary multi Template:Nowrap number is also called a unitary perfect number.
In the case k > 2, no example of a unitary multi Template:Nowrap 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.<ref name="hagis1">Template:Cite journal</ref>
The first few unitary multiply perfect numbers are:
- 1, 6, 60, 90, 87360 Template:OEIS
Bi-unitary multiply perfect numbers
A positive integer n is called a bi-unitary multi Template:Nowrap 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 Template:Nowrap number for some positive integer k.<ref name="hagis2">Template:Cite journal</ref> A bi-unitary multi Template:Nowrap number is also called a bi-unitary perfect number, and a bi-unitary multi Template:Nowrap 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.<ref name="hagis2"/>
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.<ref name="HS2020a">Template:Harvnb</ref><ref name="HS2020b">Template:Harvnb</ref><ref name="HS2020c">Template:Harvnb</ref><ref name="HS2020d">Template:Harvnb</ref><ref name="HS2021a">Template:Harvnb</ref><ref name="HS2021b">Template:Harvnb</ref>
In 2024, Tomohiro Yamada proved that 2160 is the only bi-unitary triperfect number divisible by 27 = 33.<ref>Template:Cite arXiv (published in Journal of Integer Sequences, Vol. 28 (2025), Article 25.4.5)</ref> 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 Template:OEIS
References
Sources
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See also
External links
Template:Divisor classes Template:Classes of natural numbers