A001605 - OEIS (original) (raw)

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

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

Terms n such that n-2 is also a term are listed in A279795. - M. F. Hasler, Dec 24 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(n-1) and F(n-1)*L(n+2) in some order, and F(n+2) > 1, L(n-1) > 1, F(n-1) > 1, and L(n+2) > 1 for n > 3, there are no other Fibonacci twin primes. - Bobby Jacobs, Sep 23 2017

These primes are occurring with about the same normalized frequency as Repunit primes (see Generalized Repunit Conjecture Ref). Assuming a base=1.618 (ratio of sequential terms), then the best fit coefficient is 0.60324 for the first 56 terms, which is already approaching Euler's constant 0.56145948. - Paul Bourdelais, Aug 23 2024

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, 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).

LINKS

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

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

EXTENSIONS

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