Multiply-perfect numbers

Perfect numbers, like perfect men, are very rare.—Descartes

The multiply-perfect numbers ( 

k

-perfect numbers) are the positive integers divisible by their sum of divisors.

k-perfect numbers

A  

k

-perfect number is an integer 

n

such that its sum of divisors is 

k n, k   ≥   1, k ∈ ℕ,

i.e.

$\sigma _{1}(n):=\sum _{i=1}^{\sigma _{0}(n)}d(i)=\sum _{\stackrel {i=1}{i|n}}^{n}i=\sum _{i=1}^{n}[n\,{\bmod {\,}}i=0]\cdot i=kn,\quad k\geq 1,k\in \mathbb {N} ,\,$

where 

d (i)

is the 

i

th divisor of 

n

, 

σ0(n) = τ (n)

is the number of divisors of 

n

, 

σ1(n) = σ (n)

is the sum of divisors of 

n

and 

[·]

is the Iverson bracket. Equivalently, a  

k

-perfect number is an integer 

n

such that its harmonic sum of divisors is 

k, k   ≥   1, k ∈ ℕ,

i.e.

$\sigma _{-1}(n):=\sum _{i=1}^{\sigma _{0}(n)}{\frac {1}{d(i)}}=\sum _{\stackrel {i=1}{i|n}}^{n}{\frac {1}{i}}=\sum _{i=1}^{n}[n\,{\bmod {\,}}i=0]\,\cdot \,\left({\frac {1}{i}}\right)=\sum _{i=1}^{n}[n\,{\bmod {\,}}i=0]\,\cdot \,\left({\frac {1}{n/i}}\right)=\sum _{i=1}^{n}[n\,{\bmod {\,}}i=0]\,\cdot \,\left({\frac {i}{n}}\right)={\frac {\sigma _{1}(n)}{n}}=k,\,$

where 

σ−1(n)

is the harmonic sum of divisors of 

n

. For example, 

672

is a 3-perfect number, since its divisors add up to 

2016

, and that is thrice 

672

.

Table of k-perfect numbers

For 

k   ≥   2

, it is not known whether these are finite or infinite sequences (obviously, there is only one 

1

-perfect number, i.e 

1

).

 

k

-perfect numbers

k	Sequences  ak (n), n   ≥   1.	A-number
1	{1}
2	{6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, ...}	A000396
3	{120, 672, 523776, 459818240, 1476304896, 51001180160, ...}	A005820
4	{30240, 32760, 2178540, 23569920, 45532800, 142990848, 1379454720, 43861478400, 66433720320, ...}	A027687
5	{14182439040, 31998395520, 518666803200, 13661860101120, 30823866178560, 740344994887680, 796928461056000, 212517062615531520, 69357059049509038080, ...}	A046060
6	{154345556085770649600, 9186050031556349952000, 680489641226538823680000, 6205958672455589512937472000, 13297004660164711617331200000, ...}	A046061
7	{141310897947438348259849402738485523264343544818565120000, ...}
8	{2 62 × 3 15 × 5 9 × 7 7 × 11 3 × 13 3 × 17 2 × 19 × 23 × 29 × 31 × 37 × 41 × 43 × 53 × 61 × 71 × 97 2 × 521 2 × 6118243316177221840497066178204572112368770107012542227185747, ...}

Smallest k-perfect numbers

Smallest 

k

-perfect number for each 

k   ≥   1

.
A007539 First 

n

-fold perfect number, 

n   ≥   1

.

 

{1, 6, 120, 30240, 14182439040, 154345556085770649600, 141310897947438348259849402738485523264343544818565120000, 8268099687077761372899241948635962893501943883292455548843932421413884476391773708366277840568053624227289196057256213348352000000000, ...}

1-perfect numbers

There is only one 

1

-perfect number, i.e. 

1

.

2-perfect numbers (perfect numbers)

When 

k

is not specified, it is generally understood to mean 

k = 2

, i.e. perfect numbers. These numbers have been studied since Euclid's time.

The ancient Christian scholar Augustine explained that God could have created the world in an instant but chose to do it in a perfect number of days, 6. Early Jewish commentators felt that the perfection of the universe was shown by the Moon's period of 28 days. The next in line are 496, 8128, and 33550336. All end in 6 or 8, though what seems to be an alternating pattern of 6's and 8's for the first few perfect numbers doesn't continue. In a 1638 letter to Mersenne, Descartes proposed that every even perfect number is of Euclid's form, and stated that he saw no reason why an odd perfect number could not exist (Dickson 2005, p. 12). As René Descartes pointed out: "Perfect numbers like perfect men are very rare." It isn't known if there are infinitely many perfect numbers (for each Mersenne prime we have a corresponding even perfect number, but it isn't known if there are infinitely many Mersenne primes.) It is also not known if there are any odd perfect numbers.

Even perfect numbers

The even perfect numbers are of the form

$n=2^{m}\,(2^{m+1}-1)=2^{m}\,\sigma (2^{m}),\quad m\geq 1,\,$

where 

2 m +1  −  1

must be prime (called a Mersenne prime, see A000668).

$\sigma (n)=\sigma (2^{m})\,\sigma (\sigma (2^{m}))=\sigma (2^{m})\,\sigma (2^{m+1}-1)=\sigma (2^{m})\,2^{m+1}=(2^{m+1}-1)\,2^{m+1}=2^{m+1}\,(2^{m+1}-1)=2n,\quad m\geq 1,\,$

since powers of 

2

, with positive exponent, are almost-perfect.

Theorem. (Euclid)

If 

2 m  −  1

is a prime number (called a Mersenne prime), then 

n = (2 m  − 1) (2 m  −  1)

is a perfect number (see A000396).

Proof. Labelling for our convenience the Mersenne prime as 

p = 2 m  −  1

, the divisors of 

n

are the powers of 

2

from 

1

to 

2 m  − 1

and each of those powers multiplied by 

p

. Since 

∑ m   − 1 i   = 0 2 i = 2 m  −  1

, the sum of the divisors of 

n

is 

(2 m  −  1) + (2 m  −  1) p

. Further rewriting, we get 

σ (n) = (2 m  −  1) (1 + p) = (2 m  −  1) (1 + 2 m  −  1) = 2 m (2 m  −  1) = 2 ⋅   (2 m  − 1) (2 m  −  1) = 2n

, as predicted. □

It wasn't until Leonhard Euler came along that the converse was proven.

Theorem. (Euler)

If 

n

is an even perfect number, it must be the product of a Mersenne prime 

p = 2 m  −  1

and the power of two 

2 m  − 1

. (See A000079 for the powers of 2.)

Proof. PROOF GOES HERE. □ ^{(Provide proof: PROOF GOES HERE. □)} 

Between Euclid and Euler, medieval mathematicians made some conjectures about perfect numbers that have since been proven false, such as that there is a perfect number between each consecutive power of 10 (there are no perfect numbers between 10000 and 100000, or for that matter between 10000 and 10000000), and that the least significant base 10 digit of each successive perfect number alternates between 6 and 8 (supported by reference only to the first five perfect numbers).

Every even perfect number is a triangular number, since they are a subset of

$2^{n-1}\cdot (2^{n}-1)={\frac {(2^{n}-1)2^{n}}{2}}=t_{2^{n}-1},\,$

where 

2 n  −  1

is prime.

Every even perfect number is also an hexagonal number, since they are a subset of

$2^{n-1}\cdot (2^{n}-1)=2^{n-1}\cdot (2\cdot 2^{n-1}-1)=h_{2^{n-1}},\,$

where 

2 n  −  1

is prime.

Odd perfect numbers

Main article page: Odd perfect numbers

It is not known whether odd perfect numbers exist or not! Mathematicians have been able to prove all sorts of necessary (but not sufficient) requirements for the existence of such numbers without being able to prove either that they do exist or that they don't exist.

k-perfect numbers with k ≥ 3 (multiperfect numbers)

When 

k   ≥   3

, these are considered multiperfect numbers.

Even k-perfect numbers with k ≥ 3 (even multiperfect numbers)

(...)

Odd k-perfect numbers with k ≥ 3 (odd multiperfect numbers)

(...)

Conjectured number of k-perfect numbers for each k ≥ 3

A134639 Conjectured count of numbers 

k

such that 

σ (k) k = n, n   ≥   3

.

 

{6, 36, 65, 245, 516, ...}

Follow the thread relating to the following SeqFan post on http://list.seqfan.eu/pipermail/seqfan/2012-July/thread.html#9825

---------- Forwarded message ----------
From: Georgi Guninski <guninski@guninski.com>
To: Sequence Fanatics Discussion list <seqfan@list.seqfan.eu>
Cc: 
Date: Mon, 16 Jul 2012 13:14:33 +0300
Subject: [seqfan] Re: Reference that "A027687 4-perfect numbers" is finite
Thank you.

Asked because an odd perfect number and infinitely mersenne primes implies
4-perfect numbers are infinite (and a lot of other 2k-perfect numbers) -
take the product of the OPN and coprime to it EPN.

On the other hand 4-perfect being finite and infinitely mersenne primes
implies no OPN.

What is the reason to believe all 4-perfect are discovered (even if they
are finite)?

Almost k-perfect numbers

An almost 

k

-perfect number is an integer 

n

such that its sum of divisors is 

k n  −  1, k   ≥   1, k ∈ ℕ,

i.e.

$\sigma _{1}(n):=\sum _{i=1}^{\sigma _{0}(n)}d(i)=\sum _{\stackrel {i=1}{i|n}}^{n}i=\sum _{i=1}^{n}[n\,{\bmod {\,}}i=0]\cdot i=kn-1,\quad k\geq 1,k\in \mathbb {N} ,\,$

where 

d (i)

is the 

i

th divisor of 

n

, 

σ0(n) = τ (n)

is the number of divisors of 

n

, 

σ1(n) = σ (n)

is the sum of divisors of 

n

and 

[·]

is the Iverson bracket.

Almost 1-perfect numbers

(...)

Almost 2-perfect numbers

The powers of two are almost 

2

-perfect numbers (almost-perfect numbers), since

$\sigma (2^{m})=2^{m+1}-1=2\cdot 2^{m}-1,\,m\geq 1.\,$.

Almost k-perfect numbers with k ≥ 3 (almost multiperfect numbers)

(...)

Quasi k-perfect numbers

A quasi 

k

-perfect number is an integer 

n

such that its sum of divisors is 

k n + 1, k   ≥   1, k ∈ ℕ,

i.e.

$\sigma _{1}(n):=\sum _{i=1}^{\sigma _{0}(n)}d(i)=\sum _{\stackrel {i=1}{i|n}}^{n}i=\sum _{i=1}^{n}[n\,{\bmod {\,}}i=0]\cdot i=kn+1,\quad k\geq 1,k\in \mathbb {N} ,\,$

where 

d (i)

is the 

i

th divisor of 

n

, 

σ0(n) = τ (n)

is the number of divisors of 

n

, 

σ1(n) = σ (n)

is the sum of divisors of 

n

and 

[·]

is the Iverson bracket.

Quasi 1-perfect numbers

The primes 

p

are [[quasi 

1

-perfect numbers]] since 

σ (p) = p + 1

.

Quasi 2-perfect numbers

(...)

Quasi k-perfect numbers with k ≥ 3 (quasi multiperfect numbers)

(...)

k-deficient numbers

A  

k

-deficient number is an integer 

n

such that its sum of divisors is less than 

k n, k   ≥   1, k ∈ ℕ,

i.e.

$\sigma _{1}(n):=\sum _{i=1}^{\sigma _{0}(n)}d(i)=\sum _{\stackrel {i=1}{i|n}}^{n}i=\sum _{i=1}^{n}[n\,{\bmod {\,}}i=0]\cdot i<kn,k\geq 1,k\in \mathbb {N} ,\,$

where 

d (i)

is the 

i

th divisor of 

n

, 

σ0(n) = τ (n)

is the number of divisors of 

n

, 

σ1(n) = σ (n)

is the sum of divisors of 

n

and 

[·]

is the Iverson bracket.

2-deficient numbers (deficient numbers)

Main article page: Deficient numbers

(...)

k-abundant numbers

A  

k

-abundant number is an integer 

n

such that its sum of divisors is more than 

k n, k   ≥   1, k ∈ ℕ,

i.e.

$\sigma _{1}(n):=\sum _{i=1}^{\sigma _{0}(n)}d(i)=\sum _{\stackrel {i=1}{i|n}}^{n}i=\sum _{i=1}^{n}[n\,{\bmod {\,}}i=0]\cdot i>kn,k\geq 1,k\in \mathbb {N} ,\,$

where 

d (i)

is the 

i

th divisor of 

n

, 

σ0(n) = τ (n)

is the number of divisors of 

n

, 

σ1(n) = σ (n)

is the sum of divisors of 

n

and 

[·]

is the Iverson bracket.

2-abundant numbers (abundant numbers)

Main article page: Abundant numbers

(...)

Sequences

A007691 Multiply-perfect numbers: 

n

divides 

σ (n)

.

 

{1, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, 1379454720, 1476304896, 8589869056, 14182439040, 31998395520, 43861478400, ...}

A054030 

σ (n) n

for 

n

such that 

σ (n)

is divisible by 

n

.

 

{1, 2, 2, 3, 2, 3, 2, 4, 4, 3, 4, 4, 2, 4, 4, 3, 4, 3, 2, 5, 5, 4, 3, 4, 2, 4, 4, 5, 4, 5, 5, 4, 5, 5, 4, 4, 4, 5, 4, 4, 2, 5, 4, 5, 6, 5, 5, 5, 5, 5, 5, 6, 5, 5, 4, 5, 6, 5, 4, 4, 5, 4, 5, 4, 6, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 5, 6, 5, 6, 6, 5, 4, 4, ...}

See also

• Category:Multiperfect numbers
• A007539 lists the smallest 

  k

  -perfect number.

• Perfect numbers (singles) ( 

  σ (n)  −  n = n

  )
• Amicable numbers (pairs) ( 

  σ (m)  −  m = n

  and 

  σ (n)  −  n = m

  )
• Sociable numbers ( 

  k

  -tuples) ( 

  σ (ni )  −  ni = n(i +1) mod k , i = 0 .. k  −  1, k   ≥   3

  )

References

• Dickson, L. E., History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, pp. 3-33, 2005.

External links

• The Multiply Perfect Numbers Page.
• Weisstein, Eric W., Multiperfect Number, from MathWorld—A Wolfram Web Resource.
• PlanetMath, Table of small multiply perfect numbers.
• OddPerfect.org