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A069180 F(n) and n! are relatively prime where F(n) are the Fibonacci numbers.

%I A069180 #17 Jun 07 2020 08:16:10
%S A069180 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 A069180 71,73,79,82,83,86,89,94,97,101,103,106,107,109,113,118,122,127,131,
%U A069180 134,137,139,142,146,149,151,157,163,166,167,169,173,178,179,181,191
%N A069180 F(n) and n! are relatively prime where F(n) are the Fibonacci numbers.
%C A069180 Are there any primes p >5 such that F(p) and p! are not relatively primes?
%C A069180 From _Robert Israel_, May 31 2018: (Start)
%C A069180 n is in the sequence if and only if there is no prime q = prime(k) <= n such that A001602(k) | n.
%C A069180 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 A069180 (End)
%H A069180 Robert Israel, <a href="/A069180/b069180.txt">Table of n, a(n) for n = 1..10000</a>
%F A069180 Conjecture : a(n) = C*n*Log(n) + 0(n*Log(n)) with 0, 6 < C < 0, 7
%p A069180 N:= 200: # for all terms <= N
%p A069180 V:= Vector(N,1):
%p A069180 F:= proc(n) option remember; procname(n-1)+procname(n-2) end proc:
%p A069180 F(0):= 0: F(1):= 1:
%p A069180 K:= proc(q) local k;
%p A069180    for k from 1 do if F(k) mod q = 0 then return k fi
%p A069180      od
%p A069180 end proc:
%p A069180 p:= 1:
%p A069180 do
%p A069180   p:= nextprime(p);
%p A069180   if p > N then break fi;
%p A069180   k:= K(p);
%p A069180   k0:= k*ceil(p/k);
%p A069180   V[[seq(i,i=k0..N,k)]]:= 0
%p A069180 od:
%p A069180 select(t -> V[t]=1, [$1..N]); # _Robert Israel_, May 31 2018
%t A069180 Select[Range[1000], CoprimeQ[Fibonacci[#], #!]&] (* _Jean-François Alcover_, Jun 07 2020 *)
%Y A069180 Cf. A000045, A001602.
%K A069180 easy,nonn
%O A069180 1,2
%A A069180 _Benoit Cloitre_, Apr 10 2002