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

1, 2, 7, 11, 13, 17, 19, 22, 23, 26, 29, 31, 34, 37, 41, 43, 46, 47, 53, 58, 59, 61, 62, 67, 71, 73, 79, 82, 83, 86, 89, 94, 97, 101, 103, 106, 107, 109, 113, 118, 122, 127, 131, 134, 137, 139, 142, 146, 149, 151, 157, 163, 166, 167, 169, 173, 178, 179, 181, 191

Offset: 1

Author

Benoit Cloitre, Apr 10 2002

Keywords

• easy
• nonn

Comments

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

From Robert Israel, May 31 2018: (Start)

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

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.

(End)

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000045, A001602.

Programs

Maple

N:= 200: # for all terms <= N
V:= Vector(N,1):
F:= proc(n) option remember; procname(n-1)+procname(n-2) end proc:
F(0):= 0: F(1):= 1:
K:= proc(q) local k;
   for k from 1 do if F(k) mod q = 0 then return k fi
     od
end proc:
p:= 1:
do
  p:= nextprime(p);
  if p > N then break fi;
  k:= K(p);
  k0:= k*ceil(p/k);
  V[[seq(i,i=k0..N,k)]]:= 0
od:
select(t -> V[t]=1, [$1..N]); # Robert Israel, May 31 2018

Mathematica

Select[Range[1000], CoprimeQ[Fibonacci[#], #!]&] (* Jean-François Alcover, Jun 07 2020 *)

Formula

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