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Multiplyperfect numbers
k 
Contents
 1 kperfect numbers
 2 Almost kperfect numbers
 3 Quasi kperfect numbers
 4 kdeficient numbers
 5 kabundant numbers
 6 Sequences
 7 See also
 8 Notes
 9 References
 10 External links
kperfect numbers
Ak 
n 
k n, k ≥ 1, k ∈ ℕ, 
d (i) 
i 
n 
σ0(n) = τ (n) 
n 
σ1(n) = σ (n) 
n 
[·] 
k 
n 
k, k ≥ 1, k ∈ ℕ, 
σ−1(n) 
n 
672 
2016 
672 
Table of kperfect numbers
Fork ≥ 2 
1 
1 
k 
k 
ak (n), n ≥ 1. 
{1} 
{6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, ...} 
{120, 672, 523776, 459818240, 1476304896, 51001180160, ...} 
{30240, 32760, 2178540, 23569920, 45532800, 142990848, 1379454720, 43861478400, 66433720320, ...} 
{14182439040, 31998395520, 518666803200, 13661860101120, 30823866178560, 740344994887680, 796928461056000, 212517062615531520, 69357059049509038080, ...} 
{154345556085770649600, 9186050031556349952000, 680489641226538823680000, 6205958672455589512937472000, 13297004660164711617331200000, ...} 
{141310897947438348259849402738485523264343544818565120000, ...} 
{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 kperfect numbers
Smallestk 
k ≥ 1 
A007539 First
n 
n ≥ 1 

{1, 6, 120, 30240, 14182439040, 154345556085770649600, 141310897947438348259849402738485523264343544818565120000,
8268099687077761372899241948635962893501943883292455548843932421413884476391773708366277840568053624227289196057256213348352000000000, ...}
1perfect numbers
There is only one1 
1 
2perfect numbers (perfect numbers)
Whenk 
k = 2 
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
2 m +1 − 1 
2 
Theorem. (Euclid)
Ifis a prime number (called a Mersenne prime), then
2 m − 1 is a perfect number (see A000396).
n = (2 m − 1) (2 m − 1)
Proof. Labelling for our convenience the Mersenne prime as, the divisors of
p = 2 m − 1 are the powers of
n from
2 to
1 and each of those powers multiplied by
2 m − 1 . Since
p , the sum of the divisors of
2 i = 2 m − 1
∑
m − 1
i = 0is
n . Further rewriting, we get
(2 m − 1) + (2 m − 1) p , as predicted.^{[1]} □
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
It wasn't until Leonhard Euler came along that the converse was proven.
Theorem. (Euler)
Ifis an even perfect number, it must be the product of a Mersenne prime
n and the power of two
p = 2 m − 1 . (See A000079 for the powers of 2.)
2 m − 1
Proof. PROOF GOES HERE. □ ^{(Provide proof: PROOF GOES HERE. □)} ^{[2]}
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).^{[3]}
Every even perfect number is a triangular number, since they are a subset of
2 n − 1 
Every even perfect number is also an hexagonal number, since they are a subset of
2 n − 1 
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.
kperfect numbers with k ≥ 3 (multiperfect numbers)
Whenk ≥ 3 
Even kperfect numbers with k ≥ 3 (even multiperfect numbers)
(...)
Odd kperfect numbers with k ≥ 3 (odd multiperfect numbers)
(...)
Conjectured number of kperfect numbers for each k ≥ 3
A134639 Conjectured count of numbersk 


{6, 36, 65, 245, 516, ...}
Follow the thread relating to the following SeqFan post on http://list.seqfan.eu/pipermail/seqfan/2012July/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 4perfect numbers" is finite Thank you. Asked because an odd perfect number and infinitely mersenne primes implies 4perfect numbers are infinite (and a lot of other 2kperfect numbers)  take the product of the OPN and coprime to it EPN. On the other hand 4perfect being finite and infinitely mersenne primes implies no OPN. What is the reason to believe all 4perfect are discovered (even if they are finite)?
Almost kperfect numbers
An almostk 
n 
k n − 1, k ≥ 1, k ∈ ℕ, 
d (i) 
i 
n 
σ0(n) = τ (n) 
n 
σ1(n) = σ (n) 
n 
[·] 
Almost 1perfect numbers
(...)
Almost 2perfect numbers
The powers of two are almost2 
 .
Almost kperfect numbers with k ≥ 3 (almost multiperfect numbers)
(...)
Quasi kperfect numbers
A quasik 
n 
k n + 1, k ≥ 1, k ∈ ℕ, 
d (i) 
i 
n 
σ0(n) = τ (n) 
n 
σ1(n) = σ (n) 
n 
[·] 
Quasi 1perfect numbers
The primesp 
1 
σ (p) = p + 1 
Quasi 2perfect numbers
(...)
Quasi kperfect numbers with k ≥ 3 (quasi multiperfect numbers)
(...)
kdeficient numbers
Ak 
n 
k n, k ≥ 1, k ∈ ℕ, 
d (i) 
i 
n 
σ0(n) = τ (n) 
n 
σ1(n) = σ (n) 
n 
[·] 
2deficient numbers (deficient numbers)
 Main article page: Deficient numbers
(...)
kabundant numbers
Ak 
n 
k n, k ≥ 1, k ∈ ℕ, 
d (i) 
i 
n 
σ0(n) = τ (n) 
n 
σ1(n) = σ (n) 
n 
[·] 
2abundant numbers (abundant numbers)
 Main article page: Abundant numbers
(...)
Sequences
A007691 Multiplyperfect numbers:n 
σ (n) 

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

n 
σ (n) 
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
perfect number.k
 Perfect numbers (singles) (
)σ (n) − n = n  Amicable numbers (pairs) (
andσ (m) − m = n
)σ (n) − n = m  Sociable numbers (
tuples) (k
)σ (ni ) − ni = n(i +1) mod k , i = 0 .. k − 1, k ≥ 3
Notes
References
 Dickson, L. E., History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, pp. 333, 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