superperfect number

A $k$-superperfect number $n$ is an integer such that ${\sigma }^k(n)=2n$, where ${\sigma }^k(x)$ is the iterated sum of divisors function. For example, 16 is 2-superperfect since its divisors add up to 31, and in turn the divisors of 31 add up to 32, which is twice 16.

At first Suryanarayana only considered 2-superperfect numbers. It is easy to prove that numbers of the form $2^{p-1}$ are 2-superperfect only if $2^p-1$ is a Mersenne prime. The existence of odd 2-superperfect numbers appears as unlikely as that of regular odd perfect numbers.

Later, Dieter Bode generalized the concept for any $k$ and proved that when $k>2$ there are no even $k$-superperfect numbers. Others have further generalized the concept to $(k,m)$-superperfect numbers satisifying the equality ${\sigma }^k(n)=mn$, and Weisstein programs a Mathematica command to default to $m=2$ when the third argument is omitted. For example, 8, 21, and 512 are (2, 3)-superperfect, since the second iteration of the sum of divisors function gives thrice them, 24, 63, and 1536 respectively.

Not to be confused with hyperperfect numbers, which satisfy a different equality involving the sum of divisors function.