The problem of counting certain semi-prime integers

Given two positive integers a, b, their unitary greatest common divisor is the largest divisor, d,
of a such that gcd(d, b/d) = 1. If d = 1, we say that a, b are semi-prime. The problem of counting
the number of positive integers n is discussed here. Algebraic properties of the functions involved
are established and are used to derive asymptotic estimates of their corresponding counting functions.