Superperfect Number

A number $n$ such that

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

where $\sigma(n)$ is the Divisor Function. Even superperfect numbers are just $2^{p-1}$, where $M_p=2^p-1$ is a Mersenne Prime. If any Odd superperfect numbers exist, they are Square Numbers and either $n$ or $\sigma(n)$ is Divisible by at least three distinct Primes.

More generally, an $m$-superperfect number is a number for which $\sigma^m(n)=2n$. For $m\geq 3$, there are no Even $m$-superperfect numbers.

See also Mersenne Number

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 $p^{2\alpha}$.'' Elem. Math. 24, 148-150, 1973.