Let $n\ge 4$ be an integer and $d$ the number of digits in the decimal expansion of $2^n+1$. Then $$f(n)=2^n\cdot 10^d+2^n+1$$ is the number that emerges if $2^n$ and $2^n+1$ are concatenated.
A Sophie-Germain prime is a prime $p$ with the property that $2p+1$ is prime as well.
Can $f(n)$ be a Sophie-Germain-prime ?
I neither found forced factors ( small factors, algebraic factors , etc ) nor an example of a prime $f(n)$ being a Sophie-Germain prime except the cases $n=1$ and $n=3$ , leading to the Sophie-Germain primes $23$ and $89$.