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

• Autorzy: Paul Pollack , Carl Pomerance
• Języki publikacji: EN

Abstrakty

EN

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 $x/(log x)^{1/2+o(1)}$.

Kategorie tematyczne

• 11N37: Asymptotic results on arithmetic functions
• 11N25: Distribution of integers with specified multiplicative constraints
• 11A25: Arithmetic functions; related numbers; inversion formulas

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2013
• Tom: 130
• Numer: 2
• Strony: 169-182

Daty

wydano

2013

Twórcy

autor

Paul Pollack

• Department of Mathematics, University of Georgia, Athens, GA 30602, U.S.A.

autor

Carl Pomerance

• Department of Mathematics, Dartmouth College, Hanover, NH 03755, U.S.A.

Identyfikatory

DOI

10.4064/cm130-2-3