Is there a number of the form $f(n)=7k+6=5p$ with prime $p$?

Question

Here :

Numbers $n$ of the form $10^{m}(2^{k}−1)+2^{k-1}−1$, where $m$ is the number of decimal digits of $ 2^{k-1}$.

the numbers $$f(n):=(2^n-1)\cdot 10^d+2^{n-1}-1$$ are introduced , where $d$ denotes the number of digits of $2^{n-1}-1$ in the decimal expansion. So, we simply concatenate two neighboured Mersenne-numbers, for example $f(10)=1023511$. I know no prime $f(n)$ of the form $7k+6$.

I also did not find a number $f(n)$ yet which is of the form $7k+6=5p$ with a prime $p$, so a number $f(n)$ with $f(n)\equiv 6\mod 7$ and $\frac{f(n)}{5}$ is a prime number.

With PFGW, I passed $n=122\ 000$ without finding such a number. Does such a number exist ?