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When $n$ is a prime, the additive structure of $H^*(BPU_n, \mathbb Z)$ has been computed independently in Kameko, Masaki; Yagita, Nobuaki, The Brown-Peterson cohomology of the classifying spaces of the projective unitary groups ${\rm PU}(p)$ and exceptional Lie groups. Trans. Amer. Math. Soc. 360 (2008), no. 5, 2265–2284 and in Vistoli, Angelo, On the cohomology and the Chow ring of the classifying space of ${\rm PGL}_p$. J. Reine Angew. Math. 610 (2007), 181–227. For $n = 3$, the second paper contains a computation of the multiplicative structure.