8128
nobody is perfect - numbers can be, namely: 6, 28, 496, 8128, .....
 
definition of a perfect number: sum of its divisors gives the number itself
 
for example divisors of 8128:
1,2,4,8,16,32,64,127,254,508,1016,2032,4064
1+2+4+8+16+32+64+127+254+508+1016+2032+4064 = 8128
 
it is supposed that all perfect numbers are even
 
then, for n > 6, all perfect numbers are
 
the sum of odd cubes:
28   = 1^3 + 3^3
496  = 1^3 + 3^3 + 5^3 + 7^3
8128 = 1^3 + 3^3 + 5^3 + 7^3 + 9^3 + 11^3 + 13^3 + 15^3
.....
 
and they can be written as
n    = 1 + 9*k*(k+1)/2 , k = 8j+2, j not negative integer
28   = 1 + 9*2*(2+1)/2 (j=0)
496  = 1 + 9*10*11/2   (j=1)
8128 = 1 + 9*42*43/2   (j=5)
.....
 
and all as sum of first natural numbers
6    = 1+2+3
28   = 1+2+3+4+5+6+7
496  = 1+2+3+4+5+6+7+8+ ... +31
8128 = 1+2+3+4+5+6+7+8+ ... +127
.....
 
which is the same as
6    = 3*4/2
28   = 7*8/2
496  = 31*32/2
8128 = 127*128/2
.....
 
already Euclid knew this formula:
6    = 2^1*(2^2-1)
28   = 2^1*(2^2-1)
496  = 2^4*(2^5-1)
8128 = 2^6*(2^7-1)
.....
 
reciprocals of all divisors of a perfect number n (including n) added gives 2
for 8128: 1/1+1/2+1/4+1/8+1/16+1/32+1/64+1/127+1/254+1/508+1/1016+1/2032+1/4064+1/8128 = 2
 
all perfect numbers end with 6 or 8
 
their prime factorization has the form 2^i x p, p = 2^(i+1) - 1
 
the binary code starts with k ones, followed by (k-1) zeros
 
first 7 perfect numbers (binary):

first 11 perfect numbers (decimal):

6
28
496
8 128
33 550 336
8 589 869 056
137 438 691 328
2 305 843 008 139 952 128
2 658 455 991 569 831 744 654 692 615 953 842 176
191 561 942 608 236 107 294 793 378 084 303 638 130 997 321 548 169 216
13 164 036 458 569 648 337 239 753 460 458 722 910 223 472 318 386 943 117 783 728 128
 

more interesting 4-digit numbers:  
 
more about artful numbers and their beautiful arithmetic:   artmetic