I am interested in the set $A$ of all positive integer numbers such that when factored into primes, the sum of the exponents is odd (I think of $A$ as the multiplicative odd numbers).

I want to know if it has positive upper density, more precisely $$\bar d(A):=\limsup_{n\to\infty}\frac{|A\cap[1,n]|}n$$ I think I read somewhere that it has density $1/2$ (and the $\lim$ exist, not just the $\limsup$), but I would be happy with a proof that $\bar d(A)>0$.