%I #17 Jun 07 2020 08:16:10

%S 1,2,7,11,13,17,19,22,23,26,29,31,34,37,41,43,46,47,53,58,59,61,62,67,

%T 71,73,79,82,83,86,89,94,97,101,103,106,107,109,113,118,122,127,131,

%U 134,137,139,142,146,149,151,157,163,166,167,169,173,178,179,181,191

%N F(n) and n! are relatively prime where F(n) are the Fibonacci numbers.

%C Are there any primes p >5 such that F(p) and p! are not relatively primes?

%C From _Robert Israel_, May 31 2018: (Start)

%C n is in the sequence if and only if there is no prime q = prime(k) <= n such that A001602(k) | n.

%C All primes > 5 are in the sequence, because A001602(k) < prime(k) for k > 3, and we can't have n prime unless A001602(k)=n.

%C (End)

%H Robert Israel, <a href="/A069180/b069180.txt">Table of n, a(n) for n = 1..10000</a>

%F Conjecture : a(n) = C*n*Log(n) + 0(n*Log(n)) with 0, 6 < C < 0, 7

%p N:= 200: # for all terms <= N

%p V:= Vector(N,1):

%p F:= proc(n) option remember; procname(n-1)+procname(n-2) end proc:

%p F(0):= 0: F(1):= 1:

%p K:= proc(q) local k;

%p for k from 1 do if F(k) mod q = 0 then return k fi

%p od

%p end proc:

%p p:= 1:

%p do

%p p:= nextprime(p);

%p if p > N then break fi;

%p k:= K(p);

%p k0:= k*ceil(p/k);

%p V[[seq(i,i=k0..N,k)]]:= 0

%p od:

%p select(t -> V[t]=1, [$1..N]); # _Robert Israel_, May 31 2018

%t Select[Range[1000], CoprimeQ[Fibonacci[#], #!]&] (* _Jean-François Alcover_, Jun 07 2020 *)

%Y Cf. A000045, A001602.

%K easy,nonn

%O 1,2

%A _Benoit Cloitre_, Apr 10 2002