On the distribution of some integers related to perfect and amicable numbers

Abstract

Let s′(n) = ∑ d|n, 1<d<n d be the sum of the nontrivial divisors of the natural number n, where nontrivial excludes both 1 and n. For example, s′(20) = 2 + 4 + 5 + 10 = 21. A natural number n is called quasiperfect if s′(n) = n, while n and m are said to form a quasiamicable pair if s′(n) = m and s′(m) = n; in the latter case, both n and m are called quasiamicable numbers. In this paper, we prove two statistical theorems about these classes of numbers. First, we show that the count of quasiperfect n ≤ x is at most x 1 4, as x → ∞. In fact, we show that for each fixed a, there are at most x 1 4+o(1) natural numbers n ≤ x with σ(n) ≡ a (mod n) and σ(n) odd. (Quasiperfect n satisfy these conditions with a = 1.) For fixed δ 6= 0, define the arithmetic function sδ(n) := σ(n)−n− δ. Thus, s1 = s′. Our second theorem says that the number of n ≤ x which are amicable with respect to sδ is at most x/(log x) .