Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl

PL EN

Preferencje
Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
• # Artykuł - szczegóły

## Colloquium Mathematicum

2011 | 122 | 1 | 103-123

## Powerful amicable numbers

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.

103-123

wydano
2011

### Twórcy

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