Multiply perfect number

File:Multiply perfect number Cuisenaire rods 6.png

Demonstration, with Cuisenaire rods, of the Template:Nowrap 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 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, ... (sequence A007691 in the OEIS).

ExampleEdit

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 numbersEdit

The following table gives an overview of the smallest known Template:Nowrap numbers for k ≤ 11 (sequence A007539 in the 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 × 31² × 37 × 41 × 43 × 53 × 61² × 71² × 73 × 83 × 89 × 97² × 127 × 193 × 283 × 307 × 317 × 331 × 337 × 487 × 521² × 601 × 1201 × 1279 × 2557 × 3169 × 5113 × 92737 × 649657	citation	CitationClass=web }}</ref>
9	561308081837371589999987...415685343739904000000000 (287 digits)	2¹⁰⁴ × 3⁴³ × 5⁹ × 7¹² × 11⁶ × 13⁴ × 17 × 19⁴ × 23² × 29 × 31⁴ × 37³ × 41² × 43² × 47² × 53 × 59 × 61 × 67 × 71³ × 73 × 79² × 83 × 89 × 97 × 103² × 107 × 127 × 131² × 137² × 151² × 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)	2¹⁷⁵ × 3⁶⁹ × 5²⁹ × 7¹⁸ × 11¹⁹ × 13⁸ × 17⁹ × 19⁷ × 23⁹ × 29³ × 31⁸ × 37² × 41⁴ × 43⁴ × 47⁴ × 53³ × 59 × 61⁵ × 67⁴ × 71⁴ × 73² × 79 × 83 × 89 × 97 × 101³ × 103² × 107² × 109 × 113 × 127² × 131² × 139 × 149 × 151 × 163 × 179 × 181² × 191 × 197 × 199 × 211³ × 223 × 239 × 257 × 271 × 281 × 307 × 331 × 337 × 353² × 367 × 373 × 397 × 419 × 421 × 521 × 523 × 547² × 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)	2⁴⁶⁸ × 3¹⁴⁰ × 5⁶⁶ × 7⁴⁹ × 11⁴⁰ × 13³¹ × 17¹¹ × 19¹² × 23⁹ × 29⁷ × 31¹¹ × 37⁸ × 41⁵ × 43³ × 47³ × 53⁴ × 59³ × 61² × 67⁴ × 71⁴ × 73³ × 79 × 83² × 89 × 97⁴ × 101⁴ × 103³ × 109³ × 113² × 127³ × 131³ × 137² × 139² × 149² × 151 × 157² × 163 × 167 × 173 × 181 × 191 × 193² × 197 × 199 × 211³ × 223 × 227 × 229² × 239 × 251 × 257 × 263 × 269³ × 271 × 281² × 293 × 307³ × 313 × 317 × 331 × 347 × 349 × 367 × 373 × 397 × 401 × 419 × 421 × 431 × 443² × 449 × 457 × 461 × 467 × 491 × 499² × 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/>

PropertiesEdit

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 numbersEdit

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

Tóth found several numbers that would be odd multiperfect, if one of their factors was a square (Template:Harvtxt). 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 numbers.

BoundsEdit

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 kEdit

Perfect numbersEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} A number n with σ(n) = 2n is perfect.

Triperfect numbersEdit

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 (sequence A005820 in the 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>

VariationsEdit

Unitary multiply perfect numbersEdit

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 divisor d of a number n is a unitary divisor if d and n/d share no common factors.).

A unitary multiply perfect number is simply a unitary multi Template:Nowrap number for some positive integer k. Equivalently, unitary multiply perfect numbers are those n for which n divides σ^*(n). A unitary multi Template:Nowrap number is naturally 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 more than forty four odd prime factors. This problem is probably very difficult to settle. The concept of unitary divisor was originally due to 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 (sequence A327158 in the OEIS)

Bi-unitary multiply perfect numbersEdit

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. This concept is due to Peter Hagis (1987). A bi-unitary multiply perfect number is simply a bi-unitary multi Template:Nowrap number for some positive integer k. Equivalently, bi-unitary multiply perfect numbers are those n for which n divides σ^**(n). A bi-unitary multi Template:Nowrap number is naturally called a bi-unitary perfect number, and a bi-unitary multi Template:Nowrap 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^**(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^au where 1 ≤ a ≤ 6 and u is odd,<ref name="HS2020a">Template:Harvnb</ref><ref name="HS2020b">Template:Harvnb</ref><ref name="HS2020c">Template:Harvnb</ref> and partially the case where a = 7.<ref name="HS2020d">Template:Harvnb</ref> <ref name="HS2021a">Template:Harvnb</ref> Further, they fixed completely the case a = 8.<ref name="HS2021b">Template:Harvnb</ref> Tomohiro Yamada (Determining all biunitary triperfect numbers of a certain form, arXiv:2406.19331 [math.NT], 2024) proved that 2160 = 3³ 80 is the only biunitary triperfect number of the form 3^au 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 (sequence A189000 in the OEIS)