\[n\]
is a perfect number then and \[d\]
is a divisor of \[n\]
then \[\sum_{d | n} d = 2n\]
.we can write this as
\[\sum_{d | n} \frac{n}{d}=2n\]
because as \[d\]
runs through all the divisors of \[n\]
so does \[\frac{n}{d}\]
. Hence, dividing the last expression by \[n\]
gives \[\sum_{d | n} \frac{n/d}{n}=\frac{2n}{n} \rightarrow \sum_{d | n} \frac{1}{d} = 2\]
.