number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, a superperfect number is a positive

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

that satisfies :

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

where is the sum-of-divisors function. Superperfect numbers are not a generalization of

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

s 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, ... . To illustrate: it can be seen that 16 is a superperfect number as , and , thus . If is an '' even'' superperfect number, then must be a

power of 2 A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent. In the fast-growing hierarchy, is exactly equal to f_1^n(1). In the Hardy hie ...

, , such that 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 1 ...

. It is not known whether there are any odd superperfect numbers. An odd superperfect number would have to be a

square number In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...

such that either or 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