Let $M$ be $\CP#2\CPb$, $3\CP#4\CPb$ or $(2n-1)\CP#2n\CPb$ for any integer
$n\geq 3$.

We construct an irreducible symplectic 4-manifold homeomorphic to $M$ and
also an infinite family of pairwise non-diffeomorphic irreducible
non-symplectic 4-manifolds homeomorphic to $M$. We also construct such exotic
smooth structures when $M$ is $\CP#4\CPb$ or $3\CP# k \CPb$ for $k=6,8,10$.