

A001605


Indices of prime Fibonacci numbers.
(Formerly M2309 N0911)


114



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, 2904353, 3244369, 3340367
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OFFSET

1,1


COMMENTS

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) = 13? The next candidates seem to be F(104911) or F(1968721) (greater of a pair), or F(397379), F(931517) (lesser of a pair).  M. F. Hasler, Jan 30 2013, edited Dec 24 2016, edited Sep 23 2017 by Bobby Jacobs
Henri Lifchitz confirms that the data section gives the full list (49 terms) as far as we know it today of indices of prime Fibonacci numbers (including proven primes and PRPs).  N. J. A. Sloane, Jul 09 2016
There are no Fibonacci numbers that are twin primes after F(7) = 13. Every Fibonacci prime greater than F(4) = 3 is of the form F(2*n+1). Since F(2*n+1)+2 and F(2*n+1)2 are F(n+2)*L(n1) and F(n1)*L(n+2) in some order, and F(n+2) > 1, L(n1) > 1, F(n1) > 1, and L(n+2) > 1 for n > 3, there are no other Fibonacci twin primes.  Bobby Jacobs, Sep 23 2017


REFERENCES

Clifford A. 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.
Paulo Ribenboim, The Little Book of Big Primes, SpringerVerlag, 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).


LINKS

Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 36.


FORMULA

a(n) = 1 + Sum_{m=1..L(n)} (abs(nS(m))  abs(nS(m)1/2) + 1/2), where S(m) = Sum_{k=1..m} (A010051(A000045(k))) and L(n) >= a(n)  1. L(n) can be any function of n which satisfies the inequality.  Timothy Hopper, Jun 07 2015


MATHEMATICA

Select[Range[10^4], PrimeQ[Fibonacci[#]] &] (* Harvey P. Dale, Nov 20 2012 *)
(* Start ~ 1.8x faster than the above *)
Select[Range[10^4], PrimeQ[#] && PrimeQ[Fibonacci[#]] &] (* Eric W. Weisstein, Nov 07 2017 *)
Select[Prime[Range[PrimePi[10^4]]], PrimeQ[Fibonacci[#]] &] (* Eric W. Weisstein, Nov 07 2017 *)
(* End *)


PROG

(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


CROSSREFS



KEYWORD

nonn,hard,nice


AUTHOR



EXTENSIONS

Two more terms (148091 and 201107) from T. D. Noe, Feb 12 2003 and Mar 04 2003


STATUS

approved



