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 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 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→∞.
Original language | English |
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Pages (from-to) | 3037-3046 |
Number of pages | 10 |
Journal | Journal of Number Theory |
Volume | 132 |
Issue number | 12 |
DOIs | |
State | Published - 1 Dec 2012 |
Keywords
- Perfect number
- Pseudoperfect number
- Sum-of-divisors function
ASJC Scopus subject areas
- Algebra and Number Theory