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Multiply perfect number

In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number.

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:

Smallest k-perfect numbers

The following table gives an overview of the smallest k-perfect numbers for k <= 7 (cf. Sloane's A007539):

kSmallest k-perfect numberFound by
26ancient
3120ancient
430240René Descartes, circa 1638
514182439040René Descartes, circa 1638
6154345556085770649600unknown
7141310897947438348259849402738485523264343544818565120000unknown

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