In mathematics, a superperfect number is a positive

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

''n'' that satisfies :

\sigma^2(n)=\sigma(\sigma(n))=2n\, ,

where σ is the

divisor summatory function In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

. Superperfect numbers are a generalization of

perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. T ...

s. The term was coined by D. Suryanarayana (1969). The first few superperfect numbers are : : 2, 4, 16, 64, 4096,

65536 65536 is the natural number following 65535 and preceding 65537. 65536 is a power of two: 2^ (2 to the 16th power). 65536 is the smallest number with ''exactly'' 17 divisors. In mathematics 65536 is 2^, so in tetration notation 65536 is ...

, 262144, 1073741824, ... . 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, 2^''k'', such that 2^''k''+1 − 1 is a

Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17t ...

. It is not known whether there are any

odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...

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.Guy (2004) p. 99.

Generalizations

Perfect and superperfect numbers are examples of the wider class of ''m''-superperfect numbers, which satisfy :

\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 :

\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.Guy (2007) p.79 Examples of classes of (''m'',''k'')-perfect numbers are: :

Notes

References

* * * * * {{Classes of natural numbers Divisor function Integer sequences Unsolved problems in number theory