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 (or 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 . A number that is for a certain k is called a multiply perfect number. As of 2014, numbers are known for each value of k up to 11.
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, ... .
The sum of the divisors of 120 is
1 + 2 + 3 + 4 + 5 + 6 + 8 + 10 + 12 + 15 + 20 + 24 + 30 + 40 + 60 + 120 = 360which is 3 × 120. Therefore 120 is a number.
The following table gives an overview of the smallest known numbers for k ≤ 11 :
| k | Smallest known k-perfect number | Prime factors | Found by | |
|---|---|---|---|---|
| 1 | ancient | |||
| 2 | 2 × 3 | ancient | ||
| 3 | 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 | 141310897947438348259849402738485523264343544818565120000 (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[1] | |
| 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 | |
| 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 | |
| 11 | 251850413483992918774837...000000000000000000000000 (1907 digits) | 2468 × 3140 × 566 × 749 × 1140 × 1331 × 1711 × 1912 × 239 × 297 × ... × 25922273669242462300441182317 × 15428152323948966909689390436420781 × 420391294797275951862132367930818883361 × 23735410086474640244277823338130677687887 × 628683935022908831926019116410056880219316806841500141982334538232031397827230330241 (246 distinct prime factors) | George Woltman, 2001 |
It can be proven that:
It is unknown whether there are any odd multiply perfect numbers other than 1. However if an odd number n exists where k > 2, then it must satisfy the following conditions:
In little-o notation, the number of multiply perfect numbers less than x is
o(x\varepsilon)
The number of k-perfect numbers n for n ≤ x is less than
cxc'logloglog
Under the assumption of the Riemann hypothesis, the following inequality is true for all numbers n, where k > 3
loglogn>k ⋅ e-\gamma
\gamma
The number of divisors τ(n) of a number n satisfies the inequality[2]
\tau(n)>ek.
The number of distinct prime factors ω(n) of n satisfies
\omega(n)\gek2-1.
If the distinct prime factors of n are
p1,p2,\ldots,pr
r\left(\sqrt[r]{3/2}-1\right)<
| r | |
| \sum | |
| i=1 |
| 1 | |
| pi |
<r\left(1-\sqrt[r]{6/k2}\right),~~ifniseven
r\left(\sqrt[3r]{k2}-1\right)<
| r | |
| \sum | |
| i=1 |
| 1 | |
| pi |
<r\left(1-\sqrt[r]{8/(k\pi2)}\right),~~ifnisodd
See main article: Perfect number. A number n with σ(n) = 2n is perfect.
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
If there exists an odd perfect number m (a famous open problem) then 2m would be, 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.
A similar extension can be made for unitary perfect numbers. A positive integer n is called a unitary multi 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 number for some positive integer k. Equivalently, unitary multiply perfect numbers are those n for which n divides σ*(n). A unitary multi number is naturally called a unitary perfect number. In the case k > 2, no example of a unitary multi 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
A positive integer n is called a bi-unitary multi 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 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 number is naturally called a bi-unitary perfect number, and a bi-unitary multi 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 2au where 1 ≤ a ≤ 6 and u is odd, and partially the case where a = 7.Further, they fixed completely the case a = 8.
The first few bi-unitary multiply perfect numbers are:
1, 6, 60, 90, 120, 672, 2160, 10080, 22848, 30240