Editing 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 × 31<sup>2</sup> × 37 × 41 × 43 × 53 × 61<sup>2</sup> × 71<sup>2</sup> × 73 × 83 × 89 × 97<sup>2</sup> × 127 × 193 × 283 × 307 × 317 × 331 × 337 × 487 × 521<sup>2</sup> × 601 × 1201 × 1279 × 2557 × 3169 × 5113 × 92737 × 649657 || 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 × 31<sup>4</sup> × 37<sup>3</sup> × 41<sup>2</sup> × 43<sup>2</sup> × 47<sup>2</sup> × 53 × 59 × 61 × 67 × 71<sup>3</sup> × 73 × 79<sup>2</sup> × 83 × 89 × 97 × 103<sup>2</sup> × 107 × 127 × 131<sup>2</sup> × 137<sup>2</sup> × 151<sup>2</sup> × 191 × 211 × 241 × 331 × 337 × 431 × 521 × 547 × 631 × 661 × 683 × 709 × 911 × 1093 × 1301 × 1723 × 2521 × 3067 × 3571 × 3851 × 5501 × 6829 × 6911 × 8647 × 17293 × 17351 × 29191 × 30941 × 45319 × 106681 × 110563 × 122921 × 152041 × 570461 × 16148168401 || 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> × 31<sup>8</sup> × 37<sup>2</sup> × 41<sup>4</sup> × 43<sup>4</sup> × 47<sup>4</sup> × 53<sup>3</sup> × 59 × 61<sup>5</sup> × 67<sup>4</sup> × 71<sup>4</sup> × 73<sup>2</sup> × 79 × 83 × 89 × 97 × 101<sup>3</sup> × 103<sup>2</sup> × 107<sup>2</sup> × 109 × 113 × 127<sup>2</sup> × 131<sup>2</sup> × 139 × 149 × 151 × 163 × 179 × 181<sup>2</sup> × 191 × 197 × 199 × 211<sup>3</sup> × 223 × 239 × 257 × 271 × 281 × 307 × 331 × 337 × 353<sup>2</sup> × 367 × 373 × 397 × 419 × 421 × 521 × 523 × 547<sup>2</sup> × 613 × 683 × 761 × 827 × 971 × 991 × 1093 × 1741 × 1801 × 2113 × 2221 × 2237 × 2437 × 2551 × 2851 × 3221 × 3571 × 3637 × 3833 × 4339 × 5101 × 5419 × 6577 × 6709 × 7621 × 7699 × 8269 × 8647 × 11093 × 13421 × 13441 × 14621 × 17293 × 26417 × 26881 × 31723 × 44371 × 81343 × 88741 × 114577 × 160967 × 189799 × 229153 × 292561 × 579281 × 581173 × 583367 × 1609669 × 3500201 × 119782433 × 212601841 × 2664097031 × 2931542417 × 43872038849 × 374857981681 × 4534166740403 || [[George Woltman]], 2013<ref name=fl/>
|-
| 11 || 251850413483992918774837...000000000000000000000000 (1907 digits) || 2<sup>468</sup> × 3<sup>140</sup> × 5<sup>66</sup> × 7<sup>49</sup> × 11<sup>40</sup> × 13<sup>31</sup> × 17<sup>11</sup> × 19<sup>12</sup> × 23<sup>9</sup> × 29<sup>7</sup> × 31<sup>11</sup> × 37<sup>8</sup> × 41<sup>5</sup> × 43<sup>3</sup> × 47<sup>3</sup> × 53<sup>4</sup> × 59<sup>3</sup> × 61<sup>2</sup> × 67<sup>4</sup> × 71<sup>4</sup> × 73<sup>3</sup> × 79 × 83<sup>2</sup> × 89 × 97<sup>4</sup> × 101<sup>4</sup> × 103<sup>3</sup> × 109<sup>3</sup> × 113<sup>2</sup> × 127<sup>3</sup> × 131<sup>3</sup> × 137<sup>2</sup> × 139<sup>2</sup> × 149<sup>2</sup> × 151 × 157<sup>2</sup> × 163 × 167 × 173 × 181 × 191 × 193<sup>2</sup> × 197 × 199 × 211<sup>3</sup> × 223 × 227 × 229<sup>2</sup> × 239 × 251 × 257 × 263 × 269<sup>3</sup> × 271 × 281<sup>2</sup> × 293 × 307<sup>3</sup> × 313 × 317 × 331 × 347 × 349 × 367 × 373 × 397 × 401 × 419 × 421 × 431 × 443<sup>2</sup> × 449 × 457 × 461 × 467 × 491 × 499<sup>2</sup> × 541 × 547 × 569 × 571 × 599 × 607 × 613 × 647 × 691 × 701 × 719 × 727 × 761 × 827 × 853 × 937 × 967 × 991 × 997 × 1013 × 1061 × 1087 × 1171 × 1213 × 1223 × 1231 × 1279 × 1381 × 1399 × 1433 × 1609 × 1613 × 1619 × 1723 × 1741 × 1783 × 1873 × 1933 × 1979 × 2081 × 2089 × 2221 × 2357 × 2551 × 2657 × 2671 × 2749 × 2791 × 2801 × 2803 × 3331 × 3433 × 4051 × 4177 × 4231 × 5581 × 5653 × 5839 × 6661 × 7237 × 7699 × 8081 × 8101 × 8269 × 8581 × 8941 × 10501 × 11833 × 12583 × 12941 × 13441 × 14281 × 15053 × 17929 × 19181 × 20809 × 21997 × 23063 × 23971 × 26399 × 26881 × 27061 × 28099 × 29251 × 32051 × 32059 × 32323 × 33347 × 33637 × 36373 × 38197 × 41617 × 51853 × 62011 × 67927 × 73547 × 77081 × 83233 × 92251 × 93253 × 124021 × 133387 × 141311 × 175433 × 248041 × 256471 × 262321 × 292561 × 338753 × 353641 × 441281 × 449653 × 509221 × 511801 × 540079 × 639083 × 696607 × 746023 × 922561 × 1095551 × 1401943 × 1412753 × 1428127 × 1984327 × 2556331 × 5112661 × 5714803 × 7450297 × 8334721 × 10715147 × 14091139 × 14092193 × 18739907 × 19270249 × 29866451 × 96656723 × 133338869 × 193707721 × 283763713 × 407865361 × 700116563 × 795217607 × 3035864933 × 3336809191 × 35061928679 × 143881112839 × 161969595577 × 287762225677 × 761838257287 × 840139875599 × 2031161085853 × 2454335007529 × 2765759031089 × 31280679788951 × 75364676329903 × 901563572369231 × 2169378653672701 × 4764764439424783 × 70321958644800017 × 79787519018560501 × 702022478271339803 × 1839633098314450447 × 165301473942399079669 × 604088623657497125653141 × 160014034995323841360748039 × 25922273669242462300441182317 × 15428152323948966909689390436420781 × 420391294797275951862132367930818883361 × 23735410086474640244277823338130677687887 × 628683935022908831926019116410056880219316806841500141982334538232031397827230330241 || George Woltman, 2001<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

Tóth found several numbers that would be odd multiperfect, if one of their factors was a square ({{harvtxt|Tóth|2025}}). An example is <math>8999757</math>, which would be an odd multiperfect number, if only one of its prime factors, <math>61</math>, was a square. 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 divisor ''d'' of a number ''n'' is a unitary divisor if ''d'' and ''n/d'' [[coprime integers|share no common factors]].). 

A '''unitary multiply perfect number''' is simply a unitary multi {{nowrap|''k''-perfect}} number for some positive integer ''k''. Equivalently, unitary multiply perfect numbers are those ''n'' for which ''n'' divides σ<sup>*</sup>(''n''). A unitary multi {{nowrap|2-perfect}} number is naturally 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 more than forty four odd prime factors. This problem is probably very difficult to settle. The concept of unitary divisor was originally due to [[Ramaswamy S. Vaidyanathaswamy|R. Vaidyanathaswamy]] (1931) who called such a divisor as block factor. The present terminology is due to E. Cohen (1960).

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. This concept is due to Peter Hagis (1987). A '''bi-unitary multiply perfect number''' is simply a bi-unitary multi {{nowrap|''k''-perfect}} number for some positive integer ''k''. Equivalently, bi-unitary multiply perfect numbers are those ''n'' for which ''n'' divides σ<sup>**</sup>(''n''). A bi-unitary multi {{nowrap|2-perfect}} number is naturally called a '''bi-unitary perfect number''', and a bi-unitary multi {{nowrap|3-perfect}} number is called a '''bi-unitary triperfect number'''.

A divisor ''d'' of a positive integer ''n'' is called a '''bi-unitary divisor''' of ''n'' if the greatest common unitary divisor (gcud) of ''d'' and ''n''/''d'' equals 1. This concept  is due to D. Surynarayana (1972). The sum of the (positive) bi-unitary divisors of ''n'' is denoted by σ<sup>**</sup>(''n''). 

Peter Hagis (1987) proved that there are no odd bi-unitary multiperfect numbers other than 1. Haukkanen and Sitaramaiah (2020) found all bi-unitary triperfect numbers of the form 2<sup>''a''</sup>''u'' where 1 ≤ ''a'' ≤ 6 and ''u'' is odd,<ref name="HS2020a">{{harvnb|Haukkanen|Sitaramaiah|2020a}}</ref><ref name="HS2020b">{{harvnb|Haukkanen|Sitaramaiah|2020b}}</ref><ref name="HS2020c">{{harvnb|Haukkanen|Sitaramaiah|2020c}}</ref> and partially the case where ''a'' = 7.<ref name="HS2020d">{{harvnb|Haukkanen|Sitaramaiah|2020d}}</ref>
<ref name="HS2021a">{{harvnb|Haukkanen|Sitaramaiah|2021a}}</ref>
Further, they fixed completely the case ''a'' = 8.<ref name="HS2021b">{{harvnb|Haukkanen|Sitaramaiah|2021b}}</ref>
Tomohiro Yamada (Determining all biunitary triperfect numbers of a certain form, arXiv:2406.19331 [math.NT], 2024) proved that 2160 = 3<sup>3</sup> 80 is the only biunitary triperfect number of the form 3<sup>''a''</sup>''u'' where 3 ≤ ''a''  and ''u'' is 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}}

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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]]