Do there exist prime numbers of the form $n \cdot 2^n + 1$, when $n \in \mathbb{N}$ and $n > 1$?

Question

Recently, I was studying prime sequences of the form $k \cdot 2^n + 1$, and I noticed that primes of the form $n \cdot 2^n + 1$ almost do not exist, except for the $n = 1$ case. Are there other prime numbers, or infinitely many prime numbers, when $n > 1$?

Answer 1

There's $n=141$, $n=4713$, etc. See Guy, Unsolved Problems in Number Theory, B20. They're called Cullen primes. It's not known whether there are infinitely many. A005849.

https://en.wikipedia.org/wiki/Cullen_number