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 A005478 Prime Fibonacci numbers. (Formerly M0741) 71
 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917, 475420437734698220747368027166749382927701417016557193662268716376935476241 (list; graph; refs; listen; history; text; internal format)
 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.) More generally, A190949(n) == 1 (mod 4). - N. J. A. Sloane 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 Note: A001605 gives those indices. - Antti Karttunen, 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. Brillhart, P. L. Montgomery and R. D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), 251-260. 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 N. J. A. Sloane, Table of n, a(n) for n = 1..23 C. Caldwell's The Top Twenty, Fibonacci Number. Ron Knott, Mathematics of the Fibonacci Series Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4 Eric Weisstein's World of Mathematics, Fibonacci Prime FORMULA a(n) = A000045(A001605(n)). A000040 INTERSECT A000045. - R. J. Mathar, Nov 01 2007 MATHEMATICA Select[Fibonacci[Range], PrimeQ] (* Alonso del Arte, Oct 13 2011 *) 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 Cf. A001605, A000045, A030426, A075736, A263880. Subsequence of A178762. Column k=1 of A303216. Sequence in context: A139589 A152114 A139095 * A117740 A041047 A120494 Adjacent sequences:  A005475 A005476 A005477 * A005479 A005480 A005481 KEYWORD nonn,nice,easy AUTHOR EXTENSIONS Sequence corrected by Enoch Haga, Feb 11 2000 One more term from Jason Earls, Jul 12 2001 Comment and proof added by Jonathan Sondow, May 24 2011 STATUS approved

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Last modified October 14 12:25 EDT 2019. Contains 328006 sequences. (Running on oeis4.)