Powerful amicable numbers

• Autorzy: Paul Pollack
• Języki publikacji: EN

Abstrakty

EN

Let $s(n): = ∑_{d|n, d<n} d$ denote the sum of the proper divisors of the natural number n. Two distinct positive integers n and m are said to form an amicable pair if s(n) = m and s(m) = n; in this case, both n and m are called amicable numbers. The first example of an amicable pair, known already to the ancients, is {220, 284}. We do not know if there are infinitely many amicable pairs. In the opposite direction, Erdős showed in 1955 that the set of amicable numbers has asymptotic density zero.<br> Let ℓ ≥ 1. A natural number n is said to be ℓ-full (or ℓ-powerful) if $p^{ℓ}$ divides n whenever the prime p divides n. As shown by Erdős and Szekeres in 1935, the number of ℓ-full n ≤ x is asymptotically $c_{ℓ} x^{1/ℓ}$, as x → ∞. Here $c_{ℓ}$ is a positive constant depending on ℓ. We show that for each fixed ℓ, the set of amicable ℓ-full numbers has relative density zero within the set of ℓ-full numbers.

Kategorie tematyczne

• 11N37: Asymptotic results on arithmetic functions
• 11A25: Arithmetic functions; related numbers; inversion formulas

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2011
• Tom: 122
• Numer: 1
• Strony: 103-123

Daty

wydano

2011

Twórcy

autor

Paul Pollack

• Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, U.S.A.

Identyfikatory

DOI

10.4064/cm122-1-10