  Let $PU_n$ the projective unitary group of rank $n$ and $BPU_n$ its
classifying space. For an odd prime $p$, we extend previous results to a
compete description of $H^s(BPU_n;\mathbb{Z})_{(p)}$ for $s<2p+5$ by showing
that the $p$-primary subgroups of $H^s(BPU_n;\mathbb{Z})$ are trivial for $s =
2p+3$ and $s = 2p+4$.
