What are the list of smallest integers $n$ such that $$\frac{(p_{x_{\#}})}{3} \mid n \text{ and }3^{x-1} \mid n+1?$$ Here $p_x$ denotes the $x$th prime and $p_x$# is the product of all primes less than or equal to $p_x$. of course there exists a power of $3$ such that all the primes less than or equal to $p_x$ except for $3$ divide $3^{x-1}$ but this may not be the smallest integer $n$ such that this is true for. Can someone please show me a further list than mine:
$2 \mid 2$; $3 \mid 3$
$(2\cdot 5) \mid 80$; $3^4 = 81 \mid 81$
$(2 \cdot 5 \cdot7) \mid 350$; $3^3 \mid 351$
$(2 \cdot 5 \cdot 7 \cdot 11) \mid 60830$; $3^4 \mid 60831$
$(2 \cdot 5 \cdot 7 \cdot 11 \cdot 13) \mid 310310$; $3^5 \mid 310311$
$(2 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17) \mid 92742650$; $3^6 \mid 92742651$
$(2 \cdot 5 \cdot 7\cdot 11 \cdot 13 \cdot 17 \cdot 19) \mid 5923277360$; $3^7 \mid 5923277361$
and so on.........
Thanks in advance. FYI $q = 3$ is an example. In general, find the smallest integer $n$ such that $$\frac{(p_{x_{\#}})}{p_i} \mid n \text{ and }3^{x-1} \mid n+1$$ for any prime $q$. (Here $p_x$ is the $x$th prime).