For a positive integer n, let σ(n) denote the sum of the positive divisors of n. Let d be a proper divisor of n. We call n a near-perfect number if σ(n) = 2n + d, and a deficient-perfect number if σ(n) = 2n - d. We show that there is no odd near-perfect number with three distinct prime divisors and determine all deficient-perfect numbers with at most two distinct prime factors.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm133-2-8,
author = {Min Tang and Xiao-Zhi Ren and Meng Li},
title = {On near-perfect and deficient-perfect numbers},
journal = {Colloquium Mathematicae},
volume = {131},
year = {2013},
pages = {221-226},
zbl = {1332.11004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm133-2-8}
}
Min Tang; Xiao-Zhi Ren; Meng Li. On near-perfect and deficient-perfect numbers. Colloquium Mathematicae, Tome 131 (2013) pp. 221-226. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm133-2-8/