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.
References
- 1 R. K. Guy, Unsolved Problems in Number Theory^{} New York: Springer-Verlag 2004: B9
- 2 D. Suryanarayana, “Super perfect numbers” Elem. Math. 24 (1969): 16 - 17
- 3 E. Weisstein, “http://mathworld.wolfram.com/SuperperfectNumber.htmlSuperperfect number” Mathworld
Title | superperfect number |
---|---|
Canonical name | SuperperfectNumber |
Date of creation | 2013-03-22 17:03:38 |
Last modified on | 2013-03-22 17:03:38 |
Owner | CompositeFan (12809) |
Last modified by | CompositeFan (12809) |
Numerical id | 5 |
Author | CompositeFan (12809) |
Entry type | Definition |
Classification | msc 11A25 |