A030426 a(n) = Fibonacci(prime(n)).

1, 2, 5, 13, 89, 233, 1597, 4181, 28657, 514229, 1346269, 24157817, 165580141, 433494437, 2971215073, 53316291173, 956722026041, 2504730781961, 44945570212853, 308061521170129, 806515533049393, 14472334024676221, 99194853094755497, 1779979416004714189

Offset: 1

Author

John C. Hallyburton, Jr. (jhallyburton(AT)mx1.AspenRes.Com)

Keywords

• nonn
• easy
• nice

Comments

Except for Fibonacci(4) = 3, if Fibonacci(n) is prime, then n is also prime. However, if n is prime, Fibonacci(n) might be composite, as, for example, Fibonacci(19) = 4181 = 37 * 113. - Alonso del Arte, Jan 28 2014

The values are pairwise relatively prime because gcd(Fib(m), Fib(n)) = Fib(gcd(m, n)) and this equals Fib(1) = 1 when m!=n are prime numbers. - Lee A. Newberg, May 05 2023

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..642 (first 100 terms from T. D. Noe)
• Michel Bataille, Problem 90.G, Problem Corner, The Mathematical Gazette, Vol. 90, No. 518 (2006), p. 354; Solution, ibid., Vol. 91, No. 520 (2007), pp. 160-161.

Crossrefs

Cf. A000040, A000045, A005478.

Programs

GAP

a:=List(Filtered([1..100],IsPrime),i->Fibonacci(i));; Print(a); # Muniru A Asiru, Dec 29 2018

Magma

[Fibonacci(NthPrime(n)): n in [1..80]]; // Vincenzo Librandi, May 22 2015

Maple

with(combinat); for i from 1 to 50 do fibonacci(ithprime(i)); od;
# second Maple program:
a:= n-> (<<0|1>, <1|1>>^ithprime(n))[1, 2]:
seq(a(n), n=1..30);  # Alois P. Heinz, Jan 20 2017

Mathematica

Fibonacci[Prime[Range[30]]] (* Harvey P. Dale, Mar 25 2013 *)

PARI

a(n)=fibonacci(prime(n)) \\ Charles R Greathouse IV, Apr 26 2012

Formula

a(n) = A000045(A000040(n)).

From Jianing Song, Dec 26 2018: (Start)

a(n) == 1 (mod prime(n)) if prime(n) == 1, 4 (mod 5).

a(n) == -1 (mod prime(n)) if prime(n) == 2, 3 (mod 5). (End)

a(n) == Sum_{k=0..floor((prime(n)-1)/2)} (-1)^k * binomial(2*k,k) (mod prime(n)) (Bataille, 2006). - Amiram Eldar, Jul 02 2023