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A005478
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Prime Fibonacci numbers.
(Formerly M0741)
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61
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2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917, 475420437734698220747368027166749382927701417016557193662268716376935476241
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OFFSET
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1,1
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COMMENTS
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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. - 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
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REFERENCES
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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).
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LINKS
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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
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FORMULA
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a(n) = A000045(A001605(n)). A000040 INTERSECT A000045. - R. J. Mathar, Nov 01 2007
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MATHEMATICA
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Select[Fibonacci[Range[400]], PrimeQ] (* Alonso del Arte, Oct 13 2011 *)
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PROG
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(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
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CROSSREFS
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Cf. A001605, A000045, A030426, A075736, A263880.
Subsequence of A178762.
Sequence in context: A139589 A152114 A139095 * A117740 A041047 A120494
Adjacent sequences: A005475 A005476 A005477 * A005479 A005480 A005481
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Sequence corrected by Enoch Haga, Feb 11 2000
One more term from Jason Earls (zevi_35711(AT)yahoo.com), Jul 12 2001
Comment and proof added by Jonathan Sondow, May 24 2011
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STATUS
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approved
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