Numbers n such that Fib(n-1) and Fib(n+1) are both prime.

4, 6, 12, 432, 570

1

No other terms among known Fibonacci primes. This sequence is twice S000095. Note that n-1 and n+1 are also prime.

T. D. Noe, Plot of 5 terms

(Mma) Select[Range[1000], PrimeQ[Fibonacci[# - 1]] && PrimeQ[Fibonacci[# + 1]] &]

Cf. A000045 (Fibonacci numbers), A001605 (indices of Fibonacci primes), S000095.

nonn,fini

T. D. Noe, Jun 26 2014