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a(n) = Fibonacci(prime(n)).
22

%I #60 Jul 02 2023 02:21:55

%S 1,2,5,13,89,233,1597,4181,28657,514229,1346269,24157817,165580141,

%T 433494437,2971215073,53316291173,956722026041,2504730781961,

%U 44945570212853,308061521170129,806515533049393,14472334024676221,99194853094755497,1779979416004714189

%N a(n) = Fibonacci(prime(n)).

%C 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

%C 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

%H Alois P. Heinz, <a href="/A030426/b030426.txt">Table of n, a(n) for n = 1..642</a> (first 100 terms from T. D. Noe)

%H Michel Bataille, <a href="http://www.jstor.org/stable/40378645">Problem 90.G</a>, Problem Corner, The Mathematical Gazette, Vol. 90, No. 518 (2006), p. 354; <a href="https://www.jstor.org/stable/40378316">Solution</a>, ibid., Vol. 91, No. 520 (2007), pp. 160-161.

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

%F From _Jianing Song_, Dec 26 2018: (Start)

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

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

%F 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

%p with(combinat); for i from 1 to 50 do fibonacci(ithprime(i)); od;

%p # second Maple program:

%p a:= n-> (<<0|1>, <1|1>>^ithprime(n))[1, 2]:

%p seq(a(n), n=1..30); # _Alois P. Heinz_, Jan 20 2017

%t Fibonacci[Prime[Range[30]]] (* _Harvey P. Dale_, Mar 25 2013 *)

%o (PARI) a(n)=fibonacci(prime(n)) \\ _Charles R Greathouse IV_, Apr 26 2012

%o (Magma) [Fibonacci(NthPrime(n)): n in [1..80]]; // _Vincenzo Librandi_, May 22 2015

%o (GAP) a:=List(Filtered([1..100],IsPrime),i->Fibonacci(i));; Print(a); # _Muniru A Asiru_, Dec 29 2018

%Y Cf. A000040, A000045, A005478.

%K nonn,easy,nice

%O 1,2

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