Not an answer, just some data: $$\begin{array}{|r|r|r|} \hline B_n & n & \log\log n \\ \hline -1 & 25 & 1.169 \\ -2 & 358 & 1.772 \\ -3 & 104{,}351 & 2.447 \\ 4 & 312{,}692 & 2.538 \\ 5 & 625{,}381 & 2.591 \\ 6 & 938{,}070 & 2.621 \\ -4 & 2{,}084{,}478 & 2.678 \\ -5 & 6{,}357{,}421 & 2.751 \\ -6 & 86{,}501{,}278 & 2.906 \\ -7 & 166{,}645{,}135 & 2.941 \\ 7 & 412{,}496{,}057 & 2.988 \\ 8 & 824{,}054{,}044 & 3.022 \\ 9 & 1{,}235{,}612{,}031 & 3.041 \\ 10 & 1{,}647{,}170{,}018 & 3.055 \\ 11 & 2{,}058{,}728{,}005 & 3.066 \\ 12 & 2{,}470{,}285{,}992 & 3.074 \\ -8 & 7{,}986{,}246{,}888 & 3.127 \\ \hline \end{array}$$ (I only used 50 decimal digits of precision, and I didn't check rigorously for possible errors in $$360{,}000{,}000$$ iterations, each requiring the rounding of a number to an integer. I'll try to rectify these flaws tomorrow.)