On perfect and near-perfect numbers

We call n a near-perfect number if n is the sum of all of its proper divisors, except for one of them, which we term the redundant divisor. For example, the representation12=1+2+3+6 shows that 12 is near-perfect with redundant divisor 4. Near-perfect numbers are thus a very special class of pseudoperfect numbers, as defined by Sierpiński. We discuss some rules for generating near-perfect numbers similar to Euclidʼs rule for constructing even perfect numbers, and we obtain an upper bound of x5/6+o(1) for the number of near-perfect numbers in [1,x], as x→∞.