Multiply perfect number
From Wikipedia, the free encyclopedia
Divisibility-based sets of integers |
Form of factorization: |
Prime number |
Composite number |
Powerful number |
Square-free number |
Achilles number |
Constrained divisor sums: |
Perfect number |
Almost perfect number |
Quasiperfect number |
Multiply perfect number |
Hyperperfect number |
Superperfect number |
Unitary perfect number |
Semiperfect number |
Primitive semiperfect number |
Practical number |
Numbers with many divisors: |
Abundant number |
Highly abundant number |
Superabundant number |
Colossally abundant number |
Highly composite number |
Superior highly composite number |
Other: |
Deficient number |
Weird number |
Amicable number |
Friendly number |
Sociable number |
Solitary number |
Sublime number |
Harmonic divisor number |
Frugal number |
Equidigital number |
Extravagant number |
See also: |
Divisor function |
Divisor |
Prime factor |
Factorization |
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 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 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.
[edit] Smallest k-perfect numbers
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 |
---|---|---|
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 | 141310897947438348259849402738 485523264343544818565120000 | TE Mason, 1911 |
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.