A number is said to be *perfect* if , i.e., if the sum of the proper divisors of is equal to .

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

It was known to Euclid that if is prime, then 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 .

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

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