

A005478


Prime Fibonacci numbers.
(Formerly M0741)


89



2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917, 475420437734698220747368027166749382927701417016557193662268716376935476241
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OFFSET

1,1


COMMENTS

a(n) == 1 (mod 4) for n > 2. (Proof. Otherwise 3 < a(n) = F_k == 3 (mod 4). Then k == 4 (mod 6) (see A079343 and A161553) and so k is not prime. But k is prime since F_k is prime and k != 4  see Caldwell.)
With the exception of 3, every term of this sequence has a prime index in the sequence of Fibonacci numbers (A000045); e.g., 5 is the fifth Fibonacci number, 13 is the seventh Fibonacci number, 89 the eleventh, etc.  Alonso del Arte, Aug 16 2013
The six known safe primes 2p + 1 such that p is a Fibonacci prime are in A263880; the values of p are in A155011. There are only two known Fibonacci primes p for which 2p  1 is also prime, namely, p = 2 and 3. Is there a reason for this bias toward prime 2p + 1 over 2p  1 among Fibonacci primes p?  Jonathan Sondow, Nov 04 2015


REFERENCES

J.M. De Koninck, Ces nombres qui nous fascinent, Entry 89, p. 32, Ellipses, Paris 2008.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS



FORMULA



MATHEMATICA



PROG

(PARI) je=[]; for(n=0, 400, if(isprime(fibonacci(n)), je=concat(je, fibonacci(n)))); je
(Sage) [i for i in fibonacci_xrange(0, 10^80) if is_prime(i)] # Bruno Berselli, Jun 26 2014


CROSSREFS



KEYWORD

nonn,nice,easy


AUTHOR



EXTENSIONS



STATUS

approved



