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Sun [13] studied for what values of positive integers $a,b,c$ the sum $ap_5+bp_5+cp_5$ is universal over $\\mathbb Z$ (i.e., any $n\\in\\mathbb N=\\{0,1,2,\\ldots\\}$ has the form $ap_5(x)+bp_5(y)+cp_5(z)$ with $x,y,z\\in\\mathbb Z$). We prove that $p_5+bp_5+3p_5\\,(b=1,2,3,4,9)$ and $p_5+2p_5+6p_5$ are universal over $\\mathbb Z$, as conjectured by Sun. 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