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 representation 12 = 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 Euclid’s rule for constructing even perfect numbers, and we obtain an upper bound of x for the number of near-perfect numbers in [1, x], as x→∞.