View source for Multiply perfect number

{{Short description|Number whose divisors add to a multiple of that number}}
[[File:Multiply perfect number Cuisenaire rods 6.png|thumb|Demonstration, with [[Cuisenaire rods]], of the {{nowrap|2-perfection}} 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 {{nowrap|''k''-perfect}} (or {{nowrap|''k''-fold}} perfect) if the sum of all positive [[divisor]]s of ''n'' (the [[divisor function]], ''σ''(''n'')) is equal to ''kn''; a number is thus [[perfect number|perfect]] [[if and only if]] it is {{nowrap|2-perfect}}. A number that is {{nowrap|''k''-perfect}} for a certain ''k'' is called a multiply perfect number. As of 2014, {{nowrap|''k''-perfect}} numbers are known for each value of ''k'' up to 11.<ref name=fl/>

It is unknown whether there are any [[parity (mathematics)|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, ... {{OEIS|A007691}}.

== 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 {{nowrap|3-perfect}} number.

== Smallest known ''k''-perfect numbers ==

The following table gives an overview of the smallest known {{nowrap|''k''-perfect}} numbers for ''k'' ≤ 11 {{OEIS|A007539}}:

{| class="wikitable"
! ''k'' !! Smallest ''k''-perfect number !! Factors !! Found by
|-
| 1 || [[1 (number)|1]] || ||''ancient''
|-
| 2 || [[6 (number)|6]] || 2 × 3||''ancient''
|-
| 3 || [[120 (number)|120]] || 2<sup>3</sup> × 3 × 5|| ''ancient''
|-
| 4 || 30240 || 2<sup>5</sup> × 3<sup>3</sup> × 5 × 7 || [[René Descartes]], circa 1638
|-
| 5 || 14182439040 || 2<sup>7</sup> × 3<sup>4</sup> × 5 × 7 × 11<sup>2</sup> × 17 × 19 || René Descartes, circa 1638
|-
| 6 || 154345556085770649600 (21 digits) || 2<sup>15</sup> × 3<sup>5</sup> × 5<sup>2</sup> × 7<sup>2</sup> × 11 × 13 × 17 × 19 × 31 × 43 × 257 || [[Robert Daniel Carmichael]], 1907
|-
| 7 || 141310897947438348259849...523264343544818565120000 (57 digits)|| 2<sup>32</sup> × 3<sup>11</sup> × 5<sup>4</sup> × 7<sup>5</sup> × 11<sup>2</sup> × 13<sup>2</sup> × 17 × 19<sup>3</sup> × 23 × 31 × 37 × 43 × 61 × 71 × 73 × 89 × 181 × 2141 × 599479 || TE Mason, 1911
|-
| 8 || 826809968707776137289924...057256213348352000000000 (133 digits) || 2<sup>62</sup> × 3<sup>15</sup> × 5<sup>9</sup> × 7<sup>7</sup> × 11<sup>3</sup> × 13<sup>3</sup> × 17<sup>2</sup> × 19 × 23 × 29 × ... × 487 × 521<sup>2</sup> × 601 × 1201 × 1279 × 2557 × 3169 × 5113 × 92737 × 649657 (38 distinct prime factors)|| Stephen F. Gretton, 1990<ref name=fl>{{cite web |url=http://wwwhomes.uni-bielefeld.de/achim/mpn.html |title=The Multiply Perfect Numbers Page |access-date=22 January 2014 |first=Achim |last=Flammenkamp}}</ref>
|-
| 9 || 561308081837371589999987...415685343739904000000000 (287 digits) || 2<sup>104</sup> × 3<sup>43</sup> × 5<sup>9</sup> × 7<sup>12</sup> × 11<sup>6</sup> × 13<sup>4</sup> × 17 × 19<sup>4</sup> × 23<sup>2</sup> × 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) || 2<sup>175</sup> × 3<sup>69</sup> × 5<sup>29</sup> × 7<sup>18</sup> × 11<sup>19</sup> × 13<sup>8</sup> × 17<sup>9</sup> × 19<sup>7</sup> × 23<sup>9</sup> × 29<sup>3</sup> × ... × 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) || 2<sup>413</sup> × 3<sup>145</sup> × 5<sup>73</sup> × 7<sup>49</sup> × 11<sup>27</sup> × 13<sup>22</sup> × 17<sup>11</sup> × 19<sup>13</sup> × 23<sup>10</sup> × 29<sup>9</sup> × ... × 31280679788951 × 42166482463639 × 45920153384867 × 9460375336977361 × 18977800907065531 × 79787519018560501 × 455467221769572743 × 2519545342349331183143 × 38488154120055537150068589763279 × 6113142872404227834840443898241613032969 (241 distinct prime factors)|| George Woltman, 2022<ref name=fl/>
|}

== Properties ==

It can be [[mathematical proof|proven]] that:

* For a given [[prime number]] ''p'', if ''n'' is {{nowrap|''p''-perfect}} and ''p'' does not divide ''n'', then ''pn'' is {{nowrap|(''p''&nbsp;+&thinsp;1)-perfect}}.  This implies that an [[integer]] ''n'' is a {{nowrap|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 3''n'' is {{nowrap|4''k''-perfect}} and 3 does not divide ''n'', then ''n'' is {{nowrap|3''k''-perfect}}.

== Odd multiply perfect numbers ==
{{Unsolved|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 {{nowrap|''k''-perfect}} number ''n'' exists where ''k'' > 2, then it must satisfy the following conditions:<ref name="HBI105" />
* The largest prime factor is ≥&thinsp;100129
* The second largest prime factor is ≥&thinsp;1009
* The third largest prime factor is ≥&thinsp;101

If an odd triperfect number exists, it must be greater than 10<sup>128</sup>.<ref>{{cite web |last1=Sorli |first1=Ronald |title=Algorithms in the Study of Multiperfect and Odd Perfect Numbers |url=https://opus.lib.uts.edu.au/bitstream/10453/20034/2021/02Whole.pdf |publisher=University of Technology, Sydney |access-date=11 July 2025}}</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>{{harvtxt|Tóth|2025}}</ref> This is closely related to the concept of [[Descartes number]]s.

== Bounds ==

In [[Big O notation#Little-o notation|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">{{harvnb|Sándor|Mitrinović|Crstici|2006|p=105}}</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 (mathematics)|inequality]] is true for all {{nowrap|''k''-perfect}} numbers ''n'', where ''k'' > 3
:<math>\log\log n > k\cdot e^{-\gamma}</math>
where <math>\gamma</math> is [[Euler–Mascheroni constant|Euler's gamma constant]]. This can be proven using [[Robin's theorem]].

The [[number of divisors]] τ(''n'') of a {{nowrap|''k''-perfect}} number ''n'' satisfies the inequality<ref>{{cite arXiv |last=Dagal |first=Keneth Adrian P. |eprint=1309.3527 |title=A Lower Bound for τ(n) for k-Multiperfect Number |class=math.NT |date=2013}}</ref>
:<math>\tau(n) > e^{k - \gamma}.</math>

The [[prime omega function|number of distinct prime factors]] ω(''n'') of ''n'' satisfies<ref name="HBI106">{{harvnb|Sándor|Mitrinović|Crstici|2006|p=106}}</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===
{{main|Perfect number}}
A number ''n'' with σ(''n'') = 2''n'' is '''perfect'''.

===Triperfect numbers===
A number ''n'' with σ(''n'') = 3''n'' is '''triperfect'''. There are only six known triperfect numbers and these are believed to comprise all such numbers:

: 120, 672, 523776, 459818240, 1476304896, 51001180160 {{OEIS|A005820}}

If there exists an odd perfect number ''m'' (a famous [[open problem]]) then 2''m'' would be {{nowrap|3-perfect}}, since σ(2''m'') = σ(2)σ(''m'') = 3×2''m''. An odd triperfect number must be a [[square number]] exceeding 10<sup>70</sup> and have at least 12 distinct prime factors, the largest exceeding 10<sup>5</sup>.<ref>{{harvnb|Sándor|Mitrinović|Crstici|2006|pp=108–109}}</ref>

==Variations==
===Unitary multiply perfect numbers===
A similar extension can be made for [[unitary perfect number]]s. A positive integer ''n'' is called a '''unitary multi''' {{nowrap|''k''-'''perfect'''}} '''number''' if σ<sup>*</sup>(''n'') = ''kn'' where σ<sup>*</sup>(''n'') is the sum of its [[unitary divisor]]s. A '''unitary multiply perfect number''' is a unitary multi {{nowrap|''k''-perfect}} number for some positive integer ''k''. A unitary multi {{nowrap|2-perfect}} number is also called a '''unitary perfect number'''.

In the case ''k'' > 2, no example of a unitary multi {{nowrap|''k''-perfect}} number is yet known. It is known that if such a number exists, it must be [[parity (mathematics)|even]] and greater than 10<sup>102</sup> and must have at least 45 odd prime factors.<ref name="hagis1">{{cite journal |last1=Hagis Jr. |first1=Peter |title=Lower Bounds for Unitary Multiperfect Numbers |journal=The Fibonacci Quarterly |date=1984 |volume=22 |issue=2 |pages=140–143 |doi=10.1080/00150517.1984.12429905 |url=https://www.fq.math.ca/Scanned/22-2/hagis.pdf |access-date=10 July 2025}}</ref>

The first few unitary multiply perfect numbers are:
:1, 6, 60, 90, 87360 {{OEIS|A327158}}

===Bi-unitary multiply perfect numbers===

A positive integer ''n'' is called a '''bi-unitary multi''' {{nowrap|''k''-'''perfect'''}} '''number'''  if σ<sup>**</sup>(''n'') = ''kn'' where σ<sup>**</sup>(''n'') is the sum of its [[bi-unitary divisor]]s. A '''bi-unitary multiply perfect number''' is a bi-unitary multi {{nowrap|''k''-perfect}} number for some positive integer ''k''.<ref name="hagis2">{{cite journal |last1=Hagis Jr. |first1=Peter |title=Bi-Unitary Amicable and Multiperfect Numbers |journal=The Fibonacci Quarterly |date=1987 |volume=25 |issue=2 |pages=144–150 |doi=10.1080/00150517.1987.12429711 |url=https://www.fq.math.ca/Scanned/25-2/hagis.pdf |access-date=10 July 2025}}</ref> A bi-unitary multi {{nowrap|2-perfect}} number is also called a '''bi-unitary perfect number''', and a bi-unitary multi {{nowrap|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.<ref name="hagis2"/>

In 2020, Haukkanen and Sitaramaiah studied bi-unitary triperfect numbers of the form 2<sup>''a''</sup>''u'' 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">{{harvnb|Haukkanen|Sitaramaiah|2020a}}</ref><ref name="HS2020b">{{harvnb|Haukkanen|Sitaramaiah|2020b}}</ref><ref name="HS2020c">{{harvnb|Haukkanen|Sitaramaiah|2020c}}</ref><ref name="HS2020d">{{harvnb|Haukkanen|Sitaramaiah|2020d}}</ref><ref name="HS2021a">{{harvnb|Haukkanen|Sitaramaiah|2021a}}</ref><ref name="HS2021b">{{harvnb|Haukkanen|Sitaramaiah|2021b}}</ref>

In 2024, Tomohiro Yamada proved that 2160 is the only bi-unitary triperfect number divisible by 27 = 3<sup>3</sup>.<ref>{{cite arXiv |last=Yamada |first=Tomohiro |date=2024 |title=Determining all biunitary triperfect numbers of a certain form |eprint=2406.19331 |class=math.NT}} (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 3<sup>''a''</sup>''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 {{OEIS|A189000}}

==References==
{{reflist}}

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== See also ==

* [[Hemiperfect number]]

== External links ==
* [http://wwwhomes.uni-bielefeld.de/achim/mpn.html The Multiply Perfect Numbers page]
* [http://primes.utm.edu/glossary/page.php?sort=MultiplyPerfect The Prime Glossary: Multiply perfect numbers]
* {{YouTube|id=DhPtIf-hpuU|title=The Six Triperfect Numbers}}

{{Divisor classes}}
{{Classes of natural numbers}}

[[Category:Arithmetic dynamics]]
[[Category:Divisor function]]
[[Category:Perfect numbers]]