perfect numbers

A number

is said to be perfect if $\sigma(n) = 2\cdot n$ , i.e., if the sum of the proper divisors of $n$

is equal to $n$

For example, 28 is perfect since 1 + 2 + 4 + 7 + 14 = 28.

It was known to Euclid that if $2^p-1$ is prime, then $(2^p-1)\cdot 2^{p-1}$ is a perfect number. Much time later, Euler proved that all the even perfect numbers are of this form, but it is not know if there are infinite such numbers.

It is not known if an odd perfect number may exist. However, Ochem & Rao have recently proved that such a number, if exists, must be greater than $10^{1500}$ .

The sum of the reciprocals of the divisors of a perfect number is always equal to 2.

It is easy to see that every even perfect number is also a triangular and a hexagonal number.

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

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

Useful links Mathworld, Perfect Number
Wikipedia, Perfect number
OEIS, Sequence A000396

Perfect numbers can also be... (you may click on names or numbers)