Powerful amicable numbers

Abstract

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 ` ≥ 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, 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.