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 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 2026, k-perfect numbers are known to exist 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, 459818240,14182439040, 4364925090, 8589869056 ... (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):
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:
The largest prime factor is ≥ 100129 The second largest prime factor is ≥ 1009 The third largest prime factor is ≥ 101 If an odd triperfect number exists, it must be greater than 10128. Tóth found several numbers that would be odd multiperfect, if one of their factors was a square. An example is 8999757, which would be an odd multiperfect number, if only one of its prime factors, 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 o ( x ε ) {\displaystyle o(x^{\varepsilon })} for all ε > 0. The number of k-perfect numbers n for n ≤ x is less than c x c ′ log log log x / log log x {\displaystyle cx^{c'\log \log \log x/\log \log x}} , where c and c' are constants independent of k. Under the assumption of the Riemann hypothesis, the following inequality is true for all k-perfect numbers n, where k > 3
log log n > k ⋅ e − γ {\displaystyle \log \log n>k\cdot e^{-\gamma }}
where γ {\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, where k > 2, satisfies the inequality
τ ( n ) > e k − γ . {\displaystyle \tau (n)>e^{k-\gamma }.}
The number of distinct prime factors ω(n) of n satisfies
ω ( n ) ≥ k 2 − 1 , if n is odd {\displaystyle \omega (n)\geq k^{2}-1,~~{\text{if }}n{\text{ is odd}}}
ω ( n ) ≥ k 2 / 4 , if n is even {\displaystyle \omega (n)\geq k^{2}/4,~~{\text{if }}n{\text{ is even}}}
If the distinct prime factors of n are p 1 , p 2 , … , p r {\displaystyle p_{1},p_{2},\ldots ,p_{r}} , then:
r ( 3 / 2 r − 1 ) < ∑ i = 1 r 1 p i < r ( 1 − 6 / ( k π 2 ) r ) , if n is even {\displaystyle r\left({\sqrt[{r}]{3/2}}-1\right)<\sum _{i=1}^{r}{\frac {1}{p_{i}}}<r\left(1-{\sqrt[{r}]{6/(k\pi ^{2})}}\right),~~{\text{if }}n{\text{ is even}}}
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