Infinitary harmonic numbers

The infinitary divisors of a natural number n are the products of its divisors of the , where p y is an exact prime-power divisor of n and (where y α = 0 or 1) is the binary representation of y . Infinitary harmonic numbers are those for which the infinitary divisors have integer harmonic mean. One of the results in this paper is that the number of infinitary harmonic numbers not exceeding x is less than 2.2 x 1/2 2 (1+ε)log x/log log x for any ε > 0 and x > n 0 (ε). A corollary is that the set of infinitary perfect numbers (numbers n whose proper infinitary divisors sum to n ) has density zero.