Powerful amicable numbers

Paul Pollack
2011 Colloquium Mathematicum  
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
more » ... ≥ 1. A natural number n is said to be -full (or -powerful ) if p divides n whenever the prime p divides n. As shown by Erdős and Szekeres in 1935, the number of -full n ≤ x is asymptotically c x 1/ , as x → ∞. Here c is a positive constant depending on . We show that for each fixed , the set of amicable -full numbers has relative density zero within the set of -full numbers.
doi:10.4064/cm122-1-10 fatcat:eammgi5htbgyni42exxfojq7l4