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. 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):
| k | Smallest k-perfect number | Found by |
|---|---|---|
| 1 | 1 | ancient |
| 2 | 6 | ancient |
| 3 | 120 | ancient |
| 4 | 30,240 | René Descartes, circa 1638 |
| 5 | 14,182,439,040 | René Descartes, circa 1638 |
| 6 | 154,345,556,085,770,649,600 | RD Carmichael, 1907 |
| 7 | 141,310,897,947,438,348,259,849,402,738,485,523,264,343,544,818,565,120,000 | TE Mason, 1911 |