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The fraction of $J/\psi$ mesons which come from B-meson decay,$\textrm{F}_{B{\rightarrow}J/\psi}$, is measured for J/$\psi$ rapidity\mbox{$1.2<|y|<2.2$} and $p_T>0$ in $p$+$p$ and Cu+Au collisions at$\sqrt{s_{_{NN}}}$=200 GeV with the PHENIX detector. The extracted fraction is$\textrm{F}_{B{\rightarrow}J/\psi}$ = 0.025 $\pm$ 0.006(stat) $\pm$ 0.010(syst)for $p$+$p$ collisions. For Cu+Au collisions,$\textrm{F}_{B{\rightarrow}J/\psi}$ is 0.094 $\pm$ 0.028(stat) $\pm$0.037(syst) in the Au-going direction ($-2.2<y<-1.2$) and 0.089 $\pm$0.026(stat) $\pm$ 0.040(syst) in the Cu-going direction ($1.2<y<2.2$). Thenuclear modification factor, $R_{\rm CuAu}$, of B mesons in Cu+Au collisions isconsistent with binary scaling of measured yields in $p$+$p$ at both forwardand backward rapidity.

B-meson production at forward and backward rapidity in $p$+$p$ and Cu+Au collisions at $\sqrt{s_{_{NN}}}$=200 GeV