# Sums of reciprocals of divisors, perfect numbers

Posted by Jason Polak on 08. July 2018 · Write a comment · Categories: number-theory · Tags:

A perfect number is a positive integer $n$ such that
$$\sum_{d|n} d = 2n.$$Put another way, $n$ is the sum of its proper divisors. Check out a a quick intro to perfect numbers that I wrote last November. The first three perfect numbers are $6, 28,$ and $496$. Currently, the largest perfect number, corresponding to the largest Mersenne prime is
$$(2^{77232917} – 1)\cdot 2^{77232916}$$This perfect number is over 46 million digits long!
More »