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A001605
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Indices of prime Fibonacci numbers.
(Formerly M2309 N0911)
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78
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3, 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, 2971, 4723, 5387, 9311, 9677, 14431, 25561, 30757, 35999, 37511, 50833, 81839, 104911, 130021, 148091, 201107, 397379, 433781, 590041, 593689, 604711, 931517, 1049897, 1285607, 1636007, 1803059, 1968721
(list;
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OFFSET
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1,1
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COMMENTS
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Some of the larger entries may only correspond to probable primes.
Since F[n] divides F[mn] (cf. A001578, A086597), all terms of this sequence are primes except for a(2)=4 (=2*2 but F[2]=1). - M. F. Hasler, Dec 12 2007
What is the next larger twin prime after F(4)=3, F(5)=5, F(7)=11? It seems to be >= F(104911) (greater of a pair?) or even >= F(397379) (lesser of a pair?). - M. F. Hasler, Jan 30 2013
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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.
H. Dubner and W. Keller, New Fibonacci and Lucas Primes, Math. Comp. 68 (1999) 417-427.
C. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 350.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 54.
P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, NY, 1991, p. 178.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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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Table of n, a(n) for n=1..48.
David Broadhurst, Fibonacci Numbers
David Broadhurst, Proof that F(81839) is prime, NMBRTHRY maillist, 22 April 2001
C. K. Caldwell, The Prime Glossary, Fibonacci prime
Dudley Fox, Search for Possible Fibonacci Primes
R. Knott, Mathematics of the Fibonacci Series
Henri & Renaud Lifchitz, PRP Records.
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
R. Ondrejka, The Top Ten: a Catalogue of Primal Configurations
Eric Weisstein's World of Mathematics, Fibonacci Prime.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.
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FORMULA
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Prime(i) = a(n) for some n <=> A080345(i) <= 1. - M. F. Hasler, Dec 12 2007
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MATHEMATICA
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lst={}; Do[f=Fibonacci[n]; If[PrimeQ[f], AppendTo[lst, n]], {n, 1, 10^4}]; lst [From Vladimir Joseph Stephan Orlovsky, Aug 14 2008]
Select[Range[2000000], PrimeQ[Fibonacci[#]]&] (* Harvey P. Dale, Nov 20 2012 *)
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PROG
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(PARI) v=[3, 4]; forprime(p=5, 1e5, if(ispseudoprime(fibonacci(p)), v=concat(v, p))); v \\ Charles R Greathouse IV, Feb 14 2011
(PARI) is_A001605(n)={n==4 || isprime(n) & ispseudoprime(fibonacci(n))} \\ - M. F. Hasler, Sep 29 2012
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CROSSREFS
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Cf. A005478, A000045, A001578, A086597, A080345.
Sequence in context: A140826 A081735 A107036 * A216570 A101762 A139455
Adjacent sequences: A001602 A001603 A001604 * A001606 A001607 A001608
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KEYWORD
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nonn,hard,nice,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Additional comments from Robert G. Wilson v, Aug 18 2000. More terms from David Broadhurst, Nov 08 2001. Two more terms (148091 and 201107) from T. D. Noe, Feb 12 2003 and Mar 04 2003.
397379 from T. D. Noe, Aug 18 2003
433781, 590041, 593689 from Henri Lifchitz submitted by Ray Chandler, Feb 11 2005
604711 from Henri Lifchitz communicated by Eric W. Weisstein, Nov 29 2005
931517, 1049897, 1285607 found by Henri Lifchitz circa Nov 01 2008 and submitted by Alexander Adamchuk, Nov 28 2008
1636007 from Henri Lifchitz 03/2009, communicated by Eric W. Weisstein, Apr 24 2009
1803059 and 1968721 from Henri Lifchitz 11/2009, submitted by Alex Ratushnyak, Aug 08 2012
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STATUS
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approved
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