For a given natural number k, a number n is called k-perfect (or k-fold perfect) iff the sum of all positive divisors of n (the divisor function, σ(n)) is equal to kn; a number is thus perfect iff it is 2-perfect. A number that is k-perfect for a certain k is called a multiply perfect number.
It can be proven that:
The following table gives an overview of the smallest k-perfect numbers for k <= 7 (cf. Sloane's A007539):
k | Smallest k-perfect number | Found by |
---|---|---|
2 | 6 | ancient |
3 | 120 | ancient |
4 | 30240 | René Descartes, circa 1638 |
5 | 14182439040 | René Descartes, circa 1638 |
6 | 154345556085770649600 | unknown |
7 | 141310897947438348259849402738485523264343544818565120000 | unknown |