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Powerful amicable numbers

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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.
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  • Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, U.S.A.
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bwmeta1.element.bwnjournal-article-doi-10_4064-cm122-1-10
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