From prime to perfect

The proof is in the link given for perfect numbers .

Let $P$ be a prime number in the form of $P = 2^n - 1$ , and let $x = (2^n - 1)2^{n-1} = 2^{n-1}P$ . Then the factors of $x$ are $1, 2, 4, ..., 2^{n - 1}$ and $P, 2P, 4P, ..., 2^{n - 1}P$ , which add up to $(1 + 2 + 4 + ... + 2^{n - 1}) + (P + 2P + 4P + ... + 2^{n - 1}P) = (2^n - 1) + (2^n - 1)P = (2^n - 1)(1 + P) = (2^n - 1)2^n = 2x$ . Since the factors of $x$ add up to $2x$ , $x$ is a perfect number.