Superperfect number

In mathematics a superperfect number is a positive integer "n" that satisfies

:$sigma^2(n)=sigma(sigma(n))=2n,\; ,$

where σ is the divisor function. Superperfect numbers are a generalisation of perfect numbers.

The first few superperfect numbers are

:2, 4, 16, 64, 4096, 65536, 262144 OEIS|id=A019279.

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.MathWorld|urlname=SuperperfectNumber |title=Superperfect Number ]

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 7x10²⁴. [Problem B9 in Richard K. Guy's "Unsolved Problems in Number Theory" (ISBN 0-387-94289-0)]

Perfect and superperfect numbers are examples of the wider class of ("m","k")-perfect numbers which satisfy

:$sigma^m(n)=kn,\; .$

With this notation, perfect numbers are (1,2)-perfect and superperfect numbers are (2,2)-perfect. Other classes of ("m","k")-perfect numbers are:

:

References

* [http://www-maths.swan.ac.uk/pgrads/bb/project/node34.html Superperfect Numbers] , Björn Böttcher
*PlanetMath|urlname=SuperperfectNumber|title=Superperfect Number
*G. L. Cohen and H. J. J. te Riele, "Iterating the sum-of-divisors function", Experimental Mathematics, 5 (1996), pp. 93-100