# hyperperfect number

A hyperperfect number $n$ for a given $k$ is an integer such that $k\sigma(n)=-1+k+(k+1)n$, where $\sigma(x)$ is the sum of divisors function. $n$ is then called $k$-hyperperfect. For example, 325 is 3-hyperperfect since its divisors   (1, 5, 13, 25, 65, 325) add up to 434, and $3\times 434=-1+3+(3+1)325=1302$. Numbers that are 1-hyperperfect are by default called perfect numbers, since $1\sigma(n)=-1+1+(1+1)n=2n$.

The 2-hyperperfect numbers are listed in A007593 of Sloane’s OEIS. As of 2007, the only known 3-hyperperfect number is 325. The two known 4-hyperperfect numbers are 1950625 and 1220640625, a sequence  too short to list in the OEIS, and no 5-hyperperfect numbers are known to exist. The 6-hyperperfect numbers are listed in A028499.

## References

• 1 Judson S. McCrainie, “A Study of Hyperperfect Numbers” Journal of Integer Sequences 3 (2000): 00.1.3
Title hyperperfect number HyperperfectNumber 2013-03-22 17:49:38 2013-03-22 17:49:38 CompositeFan (12809) CompositeFan (12809) 4 CompositeFan (12809) Definition msc 11A05