A Sophie Germain prime is a prime number $p$ such that $2p+1$ is also prime. What do we know about other similar "Sophie Germain-like" primes, such that for instance $4p+1$ is prime (or generally, such that $2kp+1$ is prime for $k\in\mathbb{Z}, k>1$)?
Similarly, since a succession of primes such that $p$, $2p+1$, $2(2p+1)+1$, $\ldots$ is called a Cunningham chain, are there other "Cunningham-like" chains such that $p$, $4p+1$, $4(4p+1)+1$, $\ldots$, or even $p$, $2p+1$, $4(2p+1)+1$, $8(4(2p+1)+1)+1$, $\ldots$ are primes?
From $\gcd(2p,1)=1$, according to Dirichlet's theorem on arithmetic progressions there will be infinitely many primes of such form, i.e. $2pn+1$.