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2013 | 130 | 2 | 169-182
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On the distribution of some integers related to perfect and amicable numbers

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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. <br>First, we show that the count of quasiperfect n ≤ x is at most $x^{1/4+o(1)}$ 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 ≠ 0, define the arithmetic function $s_{δ}(n) := σ(n) - n - δ$. Thus, s₁ = s'. Our second theorem says that the number of n ≤ x which are amicable with respect to $s_{δ}$ is at most $x/(log x)^{1/2+o(1)}$.
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  • Department of Mathematics, University of Georgia, Athens, GA 30602, U.S.A.
  • Department of Mathematics, Dartmouth College, Hanover, NH 03755, U.S.A.
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