Seven-segment display

I am working with the following convention for the display of the digits (note in particular the formats of $6$ , $7$ and $9$ ):

Since $17S(255) = 255$ , $18S(306) = 306$ and $20S(320) = 320$ , each of $17,18,20$ is a $7$ -segment divisor.

If $n \in \mathbb{N}$ is such that $n = 19S(n)$ , and if $n$ has $N$ digits, then $n = 19S(n) \le 19 \times 7N = 133N$ . If $N \ge 4$ then $1330N < 10^N$ , and so $n \le 133N < 10^{N-1} \le n$ , which is impossible. Thus $n$ must have at most $3$ digits. Testing the numbers from $1$ to $999$ , we find that there is indeed no integer $n \in \mathbb{N}$ such that $n = 19S(n)$ , and so $\boxed{19}$ is not a $7$ -segment divisor.