- Source: Superperfect number
In number theory, a superperfect number is a positive integer n that satisfies
σ
2
(
n
)
=
σ
(
σ
(
n
)
)
=
2
n
,
{\displaystyle \sigma ^{2}(n)=\sigma (\sigma (n))=2n\,,}
where σ is the sum-of-divisors function. Superperfect numbers are not a generalization of perfect numbers but have a common generalization. The term was coined by D. Suryanarayana (1969).
The first few superperfect numbers are :
2, 4, 16, 64, 4096, 65536, 262144, 1073741824, ... (sequence A019279 in the OEIS).
To illustrate: it can be seen that 16 is a superperfect number as σ(16) = 1 + 2 + 4 + 8 + 16 = 31, and σ(31) = 1 + 31 = 32, thus σ(σ(16)) = 32 = 2 × 16.
If n is an even superperfect number, then n must be a power of 2, 2k, such that 2k+1 − 1 is a Mersenne prime.
It is not known whether there are any odd superperfect numbers. An odd superperfect number n would have to be a square number such that either n or σ(n) is divisible by at least three distinct primes. There are no odd superperfect numbers below 7×1024.
Generalizations
Perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy
σ
m
(
n
)
=
2
n
,
{\displaystyle \sigma ^{m}(n)=2n,}
corresponding to m = 1 and 2 respectively. For m ≥ 3 there are no even m-superperfect numbers.
The m-superperfect numbers are in turn examples of (m,k)-perfect numbers which satisfy
σ
m
(
n
)
=
k
n
.
{\displaystyle \sigma ^{m}(n)=kn\,.}
With this notation, perfect numbers are (1,2)-perfect, multiperfect numbers are (1,k)-perfect, superperfect numbers are (2,2)-perfect and m-superperfect numbers are (m,2)-perfect. Examples of classes of (m,k)-perfect numbers are: