Nonaliquot numbers
YONG-GAO CHEN, QING-QING ZHAO · Publicationes Mathematicae Debrecen · 2011
For any positive integer n, let σ(n) be the sum of the positive divisors of n.It is known that almost all odd numbers can be represented in the form σ(m) -m for some natural number m.In this paper, we prove that the number of even numbers which are less than x and not of the form σ(m) -m is at least 0.06x + o(x).This improves the lower bound 1 48 x + o(x) obtained by Banks and Luca.