On the Densities of (M1, M2, M3..........)-Free Integers

Throughout this paper {p₁} denotes the sequence of primes in increasing order. That is, P₁ = 2, P₂ = 3, P₃ = 5,...... For each p₁ we associate a set M₁ of natural numbers with minimal element r, where r > 1. It is well-known that every integer m > 1 can be written uniquely as

                      m = π P₁^a₁

where all but a finite a₁ are zero.