Let $L_n$ be the $n$th Lucas number. I tested whether $L_n-1$ is prime for all $n<100000$ and found that it is prime only for $n=2,3,6,24,48,96$. Are there any other prime numbers? Also, is there a reason why most of these $n$s are in the form of $2^m\cdot3$?

We can show that $n$ must be even by these formulas for $n>3$, but they may not be useful to show that it is of the form $2^m\cdot3$: $$L_{4k}-1=L_{6k}/L_{2k}$$ $$L_{4k+1}-1=5F_{2k}\cdot F_{2k+1}$$ $$L_{4k+2}-1=F_{6k+3}/F_{2k+1}$$ $$L_{4k+3}-1=L_{2k+1}\cdot L_{2k+2}$$