Define $$f(n):=F_n+F_{2n+1}$$ where $F_n$ denotes the $n$ th fibonacci number.
Then , $f(n)$ is prime for $$n=1,4,6,12,78,348,366,432,732,864,934,1932,5700,9084,14038$$ and no other positive integer $n\le 41\ 000$
What is the next prime of this form ? Can infinite many primes of this form be expected ?
Since small prime factors are apparently not forced and considering the growth rate, I think there are infinite many primes of this form.
Best Answer
After some searching I found that $n=65070$ results in a prime (or at least a very PRP).
To this point (about 45 minutes in) I have found no other $n < 80000$ that results in a prime number.
UPDATE: A few minutes later the second new prime popped up! $n=84460$ also results in a PRP.
UPDATE: I found the third new PRP of this form, $n=110220$. Also, these findings were the only ones with $n < 115000$.
ANOTHER UPDATE: The search limit now includes all $n < 200000$ without finding additional primes.