Harmonious pairs

Abstract

Let σ be the usual sum-of-divisors function. We say that a and b form a harmonious pair if [Formula: see text]; equivalently, the harmonic mean of [Formula: see text] and [Formula: see text] is 2. For example, 4 and 12 form a harmonious pair, since [Formula: see text] and [Formula: see text]. Every amicable pair is harmonious, but there are many others. We show that the count of numbers that belong to a harmonious pair having both members in [1, x] is at most [Formula: see text], as x → ∞.