Putnam 1976 B6

B-6. As usual, let $\sigma(N)$ denote the sum of all the (positive integral) divisors of $N$. (Included among these divisors are $I$ and $N$ itself.) For example, if $p$ is a prime, then $\sigma(p)=p+1$. Motivated by the notion of a "perfect" number, a positive integer $N$ is called "quasiperfect" if $\sigma(N)=2 N+1$. Prove that every quasiperfect number is the square of an odd integer.