Kirkhart's conjecture

Is that there is some insight into the nature of primes that comes from considering the classes of numbers that share the same set of prime factors. For example

2x3x5 = 30, 4x3x5 = 60, 2x9x5 = 90, 2x3x25 = 150, 4x3x25=300 …

all of these have the same set of prime divisors {2,3,5}

in fact, each of these classes of numbers is generated by a unique square-free-number.