Multiply perfect number

In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number.

Demonstration, with Cuisenaire rods, of the 2-perfection of the number 6

For a given natural number k, a number n is called k-perfect (or k-fold 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 2-perfect. A number that is k-perfect for a certain k is called a multiply perfect number. As of 2014, k-perfect numbers are known for each value of k up to 11.^[1]

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).

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 3-perfect number.

Smallest known k-perfect numbers

The following table gives an overview of the smallest known k-perfect 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	Stephen F. Gretton, 1990[1]
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[1]
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[1]
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[1]

Properties

It can be proven that:

• For a given prime number p, if n is p-perfect and p does not divide n, then pn is (p + 1)-perfect. This implies that an integer n is a 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 3n is 4k-perfect and 3 does not divide n, then n is 3k-perfect.

Odd multiply perfect numbers

Unsolved problem in mathematics

Are there any odd multiply perfect numbers?

More unsolved problems in mathematics

It is unknown whether there are any odd multiply perfect numbers other than 1. However if an odd k-perfect number n exists where k > 2, then it must satisfy the following conditions:^[2]

• 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 (Tóth (2025)). An example is ${\displaystyle 8999757}$, which would be an odd multiperfect number, if only one of its prime factors, ${\displaystyle 61}$, 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 ${\displaystyle o(x^{\varepsilon })}$ for all ε > 0.^[2]

The number of k-perfect numbers n for n ≤ x is less than ${\displaystyle cx^{c'\log \log \log x/\log \log x}}$, where c and c' are constants independent of k.^[2]

Under the assumption of the Riemann hypothesis, the following inequality is true for all k-perfect numbers n, where k > 3

${\displaystyle \log \log n>k\cdot e^{-\gamma }}$

where ${\displaystyle \gamma }$ is Euler's gamma constant. This can be proven using Robin's theorem.

The number of divisors τ(n) of a k-perfect number n satisfies the inequality^[3]

${\displaystyle \tau (n)>e^{k-\gamma }.}$

The number of distinct prime factors ω(n) of n satisfies^[4]

${\displaystyle \omega (n)\geq k^{2}-1.}$

If the distinct prime factors of n are ${\displaystyle p_{1},p_{2},\ldots ,p_{r}}$, then:^[4]

${\displaystyle r\left({\sqrt[{r}]{3/2}}-1\right)<\sum _{i=1}^{r}{\frac {1}{p_{i}}}<r\left(1-{\sqrt[{r}]{6/k^{2}}}\right),~~{\text{if }}n{\text{ is even}}}$

${\displaystyle r\left({\sqrt[{3r}]{k^{2}}}-1\right)<\sum _{i=1}^{r}{\frac {1}{p_{i}}}<r\left(1-{\sqrt[{r}]{8/(k\pi ^{2})}}\right),~~{\text{if }}n{\text{ is odd}}}$

Specific values of k

Perfect numbers

Main article: Perfect number

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 (sequence A005820 in the OEIS)

If there exists an odd perfect number m (a famous open problem) then 2m would be 3-perfect, 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.^[5]

Variations

Unitary multiply perfect numbers

A similar extension can be made for unitary perfect numbers. A positive integer n is called a unitary multi k-perfect 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 k-perfect number for some positive integer k. Equivalently, unitary multiply perfect numbers are those n for which n divides σ^*(n). A unitary multi 2-perfect number is naturally called a unitary perfect number. In the case k > 2, no example of a unitary multi k-perfect 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 numbers

A positive integer n is called a bi-unitary multi k-perfect 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 k-perfect 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 2-perfect number is naturally called a bi-unitary perfect number, and a bi-unitary multi 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^**(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,^[6][7][8] and partially the case where a = 7.^{[9] [10]} Further, they fixed completely the case a = 8.^[11] 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)