Perfect number

A perfect number is an integer that is equal to the sum of its divisors. excluding the number itself.

The first even perfect numbers are: 6, 28, 496, 8128, 33550336, 8589869056, 137438691328...

Perfect numbers are also divisible by Mersenne primes.

All perfect numbers are proven to be triangular, hexagonal, pernicious, and even.

Odd perfect numbers
It is unknown where there are any odd perfect numbers, But it must satisfy these conditions:
 * It is greater than 10^1500.
 * It isn't divisible by 105.
 * It is of the form of N = 1 (mod 12) or N = 117 (mod 468), or N = 81 (mod 324)
 * It's largest prime factor is larger than 100,000,000, And less than (3n)^1/3.
 * It's second largest prime factor is larger than 10,000, And less than (2n)^1/5.
 * It's third-largest factor is larger than 100.

Triperfect numbers
A triperfect number is an integer that has an aliquot sum of 3n, including the number itself. Where n is a triperfect number.

The first triperfect numbers are: 120, 672, 523776, 459818240, 1476304896, 51001180160. It is proven that there are only 6.

There may be infinite, or finite triperfect numbers.

It is stated that, If n is a odd perfect number, Then 2n is a triperfect number.

Other multiply perfect numbers
A k-perfect number is an integer that has an aliquot sum of kn, including the number itself. Where is a triperfect number.

Here is a table, Listing the first few terms of k-perfect numbers.