Following Terry's method and GH from MO, I started from the upper bound of GH from MO and found this pretty quick: $n=4483$ $$ \{p_1,\dots,p_{13}\}=\{67,71,73,79,83,89,97,101,103,107,109,113,127\}, $$$$ \{p_1,\dots,p_{13}\}=\{67,71,73,79,83,89,97,101,103,107,109,113,127\} $$
$$ \{q_1,\dots,q_7\}=\{4349,4441,4447,4451,4457,4481,4483\}. $$$$ \{q_1,\dots,q_7\}=\{4349,4441,4447,4451,4457,4481,4483\} $$
$$ \sum_{m=1}^{7}\log q_m-\sum_{j=1}^{13}\log p_j = 1.66862917368\times 10^{-5}, $$
$$ \sum_{j=1}^{13}\log\!\left(1+\frac{1}{p_j^2+p_j}\right) -\sum_{m=1}^{7}\log\!\left(1+\frac{1}{q_m}\right) = 5.62404029424\times 10^{-5}. $$