A099000 Indices k such that the k-th prime is a Fibonacci number.

1, 2, 3, 6, 24, 51, 251, 3121, 42613, 23023556, 143130479, 2602986161967491

Offset: 1

Author

Rick L. Shepherd, Nov 06 2004

Keywords

• nonn
• hard
• more

Comments

From Hugo Pfoertner, Jan 06 2020: (Start)

The computation of the next two terms, corresponding to the primes F(131) = A005478(13) = 1066340417491710595814572169, and F(137) = A005478(14) = 19134702400093278081449423917, should already be within reach with current (2020) technology, e.g. with Kim Walisch's "primecount" program, which allows massive parallelization. An exact determination of the following term a(15), which corresponds to F(359), is beyond any imaginable technical possibility.

Estimates for a(13)-a(15), found by using the PARI program from A121046 in a bisection loop, with an accuracy that corresponds to the shown number of digits, are as follows:

a(13) = primepi(F(131)) ~= 1.741898800848...*10^25,

a(14) = primepi(F(137)) ~= 2.9848914766265...*10^26,

a(15) = primepi(F(359)) ~= 2.78114064956041656819790214151422895...*10^72.

(End)

Crossrefs

Cf. A001605 (n-th Fibonacci number is prime), A005478 (Prime Fibonacci numbers).

Cf. A121046.

Programs

Mathematica

PrimePi[Select[Fibonacci[Range[80]], PrimeQ]]

PARI

print1("1, 2");forprime(p=5,47,if(isprime(fibonacci(p)),print1(", "primepi(fibonacci(p))))) \\ Charles R Greathouse IV, Aug 21 2011

Formula

a(n) = A000720(A005478(n)). - M. F. Hasler, Aug 21 2011

Extensions

a(11) from Ryan Propper, Oct 16 2005

a(12) from Charles R Greathouse IV, Aug 21 2011