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Powerful amicable numbers

2011
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Colloquium Mathematicum
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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. Let

doi:10.4064/cm122-1-10
fatcat:eammgi5htbgyni42exxfojq7l4