A perfect number is a number whose
excluding the number itself, add up to the number. 6 is a perfect number. The divisors
of 6 are 1, 2, 3, and 6. Leaving the 6 out, the sum is 1+2+3=6. The second perfect number
is 28: 1+2+4+7+14=28). The third perfect number is 496: 1+2+4+8+16+31+62+124+248. The
next few perfect numbers are 8128, 33550336, 8589869056, 137438691328, 2305843008139952128.
Properties of Perfect Numbers
Not a perfect number
It is not known if any odd perfect numbers exist. All odd numbers up to 10300
have been checked without finding a perfect number.
Even perfect numbers greater
than 6 take the form
where Tn is the triangular number
where j is a natural number. However not all numbers generated by
this algorithm are perfect numbers. See table 1.
All even perfect numbers are also hexagonal numbers that are the sum of a set of
consecutive integers starting at 1. An example is 28 = 1+2+3+4+5+6+7. See table 1.
The sum of the reciprocals of the divisors all the even perfect numbers is 2.
Generate Perfect Numbers
This sections uses a program to generate perfect numbers. Click on the 'Generate'
button to check one value to see if it is a perfect number. Click again to see the next,
and so on. The results of the generation will appear below.