A297624 Numbers k such that Fibonacci(2*k+1) and Fibonacci(2*k-1) are prime.

2, 3, 6, 216, 285

Offset: 1

Author

Vincenzo Librandi, Jan 08 2018

Keywords

• nonn
• more

Comments

a(6) > 1622184 if it exists (see A001605). - Chai Wah Wu, Jan 23 2018

Examples

			2 is in the sequence because F(3)=2 and F(5)=5 are prime.
6 is in the sequence because F(11)=89 and F(13)=233 are prime.
		

Crossrefs

Cf. A000045, A001605, A073340, A117517, A117595, A279795, A281087.

Programs

GAP

o := [];; for k in [1..500] do if IsPrime(Fibonacci(2*k+1)) and IsPrime(Fibonacci(2*k-1)) then Add(o,k); fi; od; A297624 := o; # Muniru A Asiru, Jan 25 2018

Magma

[n: n in [0..700] | IsPrime(Fibonacci(2*n+1)) and  IsPrime(Fibonacci(2*n-1))];

Maple

with(combinat, fibonacci): select(k -> isprime(fibonacci(2*k+1)) and isprime(fibonacci(2*k-1)), [$1..500]); # Muniru A Asiru, Jan 25 2018

Mathematica

Select[Range[0, 3000], PrimeQ[Fibonacci[2 # + 1]] && PrimeQ[Fibonacci[2 # - 1]] &]

PARI

isok(n) = isprime(fibonacci(2*n-1)) && isprime(fibonacci(2*n+1)); \\ Michel Marcus, Jan 08 2018

Python

from sympy import isprime
A297624_list, k, a, b, c, aflag = [], 1, 1, 1, 2, False
while k < 1000:
    cflag = isprime(c)
    if aflag and cflag:
        A297624_list.append(k)
    k, a, b, c, aflag = k + 1, c, b + c, b + 2*c, cflag # Chai Wah Wu, Jan 23 2018

Formula

From Chai Wah Wu, Jan 23 2018: (Start)

a(n) = (A279795(n)-1)/2 = (A281087(n)+1)/2 = (A073340(2n-1)+1)/2.

For n > 1, a(n) == 0 mod 3 as otherwise Fibonacci(2*k+1) or Fibonacci(2*k-1) is even. (End)