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 known 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 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[1]
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[1]
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[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

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

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 nx 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}}}
${\displaystyle r\left({\sqrt[{3r}]{k^{2}}}-1\right)<\sum _{i=1}^{r}{\frac {1}{p_{i}}}

Specific values of k

Perfect numbers

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 2au 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]

The first few bi-unitary multiply perfect numbers are:

1, 6, 60, 90, 120, 672, 2160, 10080, 22848, 30240 (sequence A189000 in the OEIS)

References

1. Flammenkamp, Achim. "The Multiply Perfect Numbers Page". Retrieved 22 January 2014.
2. ^ a b c
3. ^ Dagal, Keneth Adrian P. (2013). "A Lower Bound for τ(n) for k-Multiperfect Number". arXiv:1309.3527 [math.NT].
4. ^ a b
5. ^ Sándor, Mitrinović & Crstici 2006, pp. 108–109
6. ^ Haukkanen & Sitaramaiah 2020a
7. ^ Haukkanen & Sitaramaiah 2020b
8. ^ Haukkanen & Sitaramaiah 2020c
9. ^ Haukkanen & Sitaramaiah 2020d
10. ^ Haukkanen & Sitaramaiah 2021a
11. ^ Haukkanen & Sitaramaiah 2021b