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More precisely, for each admissible pair $(m, k)$, we obtain an explicit formula for $v_3(s(a3^n, a3^m-k))$. The proof combines properties of the $m$-th Stirling numbers of the first kind with a detailed analysis of the relevant $3$-adic orders. As a consequence, we prove the case $p=3$ of a conjecture of Hong and Qiu proposed in 2020. 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