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perfect numbers
A number  $n$  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
remainders of perfect numbers
Useful links Mathworld, Perfect Number
Wikipedia, Perfect number
OEIS, Sequence A000396
Perfect numbers can also be... (you may click on names or numbers)

aban 28 496 alternating 496 amenable 28 496 8128 33550336 apocalyptic 8128 binomial 28 496 8128 33550336 8589869056 137438691328 c.nonagonal 28 496 8128 33550336 8589869056 137438691328 congruent 28 496 8128 Cunningham 28 fibodiv 28 frugal 33550336 happy 28 496 8128 harmonic 28 496 8128 33550336 8589869056 137438691328 Harshad 8589869056 hexagonal 28 496 8128 33550336 8589869056 137438691328 hyperperfect 28 496 8128 33550336 8589869056 137438691328 idoneal 28 metadrome 28 nude 8128 oban 28 odious 28 496 8128 33550336 pernicious 28 496 8128 plaindrome 28 practical 28 496 8128 pseudoperfect 28 496 8128 33550336 repfigit 28 triangular 28 496 8128 33550336 8589869056 137438691328 uban 28 Ulam 28 8128 upside-down 28 wasteful 28 496 8128 Zumkeller 28 496 8128