On perfect and near-perfect numbers

Abstract

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 representationshows that 12 is near-perfect with redundant divisor 4. Nearperfect 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 5/6+o(1) for the number of near-perfect numbers in [1, x], as x → ∞.

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“…Somewhat disappointingly, despite the fact that our third theorem was new when we proved it, a proof appeared in a paper by Pollack and Shevelev [2012] after our work was submitted to Involve. We discovered this work while reading references recommended by the referee during revisions.…”

“…is studied in [Pollack and Shevelev 2012]. These are integers whose excedent is equal to one of the divisors.…”

A Class of New Near-Perfect Numbers

“…Until now, the conjecture has not been proved. Therefore, people study other similar numbers, such as pseudoperfect numbers, near-perfect numbers, k-near-perfect numbers and deficient-perfect numbers, which are closely related to perfect numbers (see [8,9,10,13]). …”

“…In 2012, based on the criterion for the existence of even perfect numbers, Paul Pollack and Vladimir Shevelev obtained 3 classes of even near-perfect numbers as follows (see [8]). Proposition 1.2.…”

“…Beyond that, in [8], the authors generalized near-perfect numbers to k-nearperfect numbers and proved the following proposition.…”