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. As of July 2004, k-perfect numbers are known for each value of k up to 11.

It can be proven that:

• For a given prime number p, if n is p-perfect and p does not divide n, then pn is (p+1)-perfect. This implies that if an integer n is a 3-perfect number divisible by 2 but not by 4, then n/2 is an odd perfect number, of which none are known.
• If 3n is 4k-perfect and 3 does not divide n, then n is 3k-perfect.

Smallest k-perfect numbers

The following table gives an overview of the smallest k-perfect numbers for k <= 7 (cf. Sloane's A007539 (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A007539)):

External links

• The Multiply Perfect Numbers page (http://wwwhomes.uni-bielefeld.de/achim/mpn.html)
• The Prime Glossary: Multiply perfect numbers (http://primes.utm.edu/glossary/page.php?sort=MultiplyPerfect)