## Superperfect Number

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.