Consider $X=\{q(q^n-1)|q \ \text{is some power of a prime number}, n\in \Bbb N^*\}$, $S=\{ s \in \Bbb N| s=\prod_i s_i, s_i \in X\} $, I am interested in which integers are in $S$. For example, $2, 6 \in S$ while $4,5 \notin S$.

1. Does $S$ have a positive density?
2. Does there exist arbitrary long consecutive sequences in $S$?

Consider $X=\{q(q^n-1)|q \ \text{is some power of a prime number}, n\in \Bbb N^*\}$, $S=\{ s \in \Bbb N| s=\prod_i s_i, s_i \in X\} $, I am interested in which integers are in $S$. For example, $2, 6 \in S$ while $5 \notin S$.

1. Does $S$ have a positive density?
2. Does there exist arbitrary long consecutive sequences in $S$?