If as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, the product will be perfect.
Proof
Let s=1+2+4+⋯+2n−1=2n−1 be prime (a Mersenne
prime); set N=s⋅2n−1. The proper divisors of N are
1,2,4,…,2n−1,s,2s,…,2n−2s, whose sum is
(2n−1)+s(2n−1−1)=s+s(2n−1−1)=s⋅2n−1=N. So N equals the sum of its proper divisors and is perfect.