We know more about divisor than non-divisors, If we consider the sets :
$$P^{1}_{x} =\left \{ p \leq x : \ p \in \mathbb{P} \right \}$$ ($\mathbb{P} $ is the primes set)
$$ P^{2}_{x} =\left \{ p_{i} \times p_{j}\ : \left( p_{i},p_{j} \right) \in \left(P^{1}_{x} \right)^{2} ,\ i < j \right \} $$
and in general the set of products with no prime repetition :
$$ P^{n}_{x} =\left \{ p_{i} \times .. \times p_{n}\ : \left( p_{i},p_{j} \right) \in \left(P^{1}_{x} \right)^{n} ,\ i <..< n \right \} $$
And we define :
$$ G^{n}_{x} =\left \{ p \in P^{n}_{x} \ : p \not| x \right \} $$ The set of primes and product of prime (with no repetition) that does NOT devide $x$.
It's easy to prove that ( where $p_{l}$ is the last prime $ \leq x $ ):
$$ Card(G^{1}_{x}) - Card(G^{2}_{x}) + Card(G^{3}_{x})-..+(-1)^{l+1} Card(G^{l}_{x}) = 0 $$
Do you know any non trivial bounds for the non alterned sum :
$$ Card(G^{1}_{x}) + Card(G^{2}_{x}) + Card(G^{3}_{x})+..+ Card(G^{l}_{x}) $$