A number such that

where is the Divisor Function. Even superperfect numbers are just , where is a Mersenne Prime. If any Odd superperfect numbers exist, they are Square Numbers and either or is Divisible by at least three distinct Primes.

More generally, an -superperfect number is a number for which . For , there are no Even -superperfect numbers.

**References**

Guy, R. K. ``Superperfect Numbers.'' §B9 in
*Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag, pp. 65-66, 1994.

Kanold, H.-J. ``Über `Super Perfect Numbers.''' *Elem. Math.* **24**, 61-62, 1969.

Lord, G. ``Even Perfect and Superperfect Numbers.'' *Elem. Math.* **30**, 87-88, 1975.

Suryanarayana, D. ``Super Perfect Numbers.'' *Elem. Math.* **20**, 16-17, 1969.

Suryanarayana, D. ``There is No Odd Super Perfect Number of the Form .'' *Elem. Math.* **24**, 148-150, 1973.

© 1996-9

1999-05-26