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) if and only if the sum of all positive divisors of n (the divisor function, σ(n)) is equal to kn; a number is thus perfect if and only if it is 2-perfect. A number that is k-perfect for a certain k is called a multiply perfect number. As of 2014, 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 an integer n is a 3-perfect number divisible by 2 but not by 4, if and only if 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 ≤ 11 (sequence A007539 in OEIS):

k	Smallest k-perfect number	Found by
1	1	ancient
2	6	ancient
3	120	ancient
4	30240	René Descartes, circa 1638
5	14182439040	René Descartes, circa 1638
6	154345556085770649600	Robert Daniel Carmichael, 1907
7	141310897947438348259849402738485523264343544818565120000	TE Mason, 1911
8	2.34111439263306338... × 10161	Paul Poulet, 1929
9	7.9842491755534198... × 10465	Fred Helenius
10	2.86879876441793479... × 10923	Ron Sorli
11	2.51850413483992918... × 101906	George Woltman

For example, 120 is 3-perfect because the sum of the divisors of 120 is
1+2+3+4+5+6+8+10+12+15+20+24+30+40+60+120 = 360 = 3 × 120.

Properties

• The number of multiperfect numbers less than X is $o(X^{\epsilon})$ for all positive ε.
• The only known odd multiply perfect number is 1.

Specific values of k

Perfect numbers

Main article: Perfect number

A number n with σ(n) = 2n is perfect.

Triperfect numbers

A number n with σ(n) = 3n is triperfect. An odd triperfect number must exceed 10⁷⁰, have at least 12 distinct prime factors, the largest exceeding 10⁵.

External links

• The Multiply Perfect Numbers page
• The Prime Glossary: Multiply perfect numbers