I am trying to prove the following theorem:
Theorem. A number is perfect iff the sum of the reciprocal of its divisors, excluding $1$, is $1$.
Thus far, this is the proof that I have managed to sew:
Proof. Let $n$ be perfect. Then $2n=1+a_1+a_2+\cdots+a_m+n$, where each $a_j$ is a divisor of $n$. Moreover, let $1 Nevertheless, I do not feel very confident about it. What do you guys think?