%I #52 Oct 04 2021 16:18:22

%S 1,2,3,4,5,7,8,11,13,17,18,21,29,34,47,48,55,72,76,89,122,123,144,199,

%T 233,305,322,323,329,377,521,610,842,843,987,1292,1353,1364,1597,2207,

%U 2208,2255,2584,3571,4181,5473,5777,5778,5796,6765,9349

%N Integer ratios of Fibonacci numbers F(m)/F(n).

%C Phong shows that no member of this sequence is perfect. - _Charles R Greathouse IV_, Jul 26 2011

%C Every number in the sequence except 1 can be written uniquely as the quotient of two Fibonacci numbers. - _M. Farrokhi D. G._, Jul 24 2020

%C From the Binet's formula for the n-th Fibonacci number, F(n), it can be demonstrated that F(n) / F(d) can be an integer only if d is a divisor of n. See also the M. Farrokhi D. G. link. - _Robert G. Wilson v_, Sep 22 2021

%H T. D. Noe, <a href="/A031121/b031121.txt">Table of n, a(n) for n = 1..1000</a>

%H M. Farrokhi D. G., <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Farrokhi/farrokhi20.html">Some remarks on the equation F_n=kF_m in Fibonacci numbers</a>, J. Integer Seq. 10 (2007), no. 5, Article 07.5.7, 9 pp.

%H Florian Luca and V. Janitzio Mejía Huguet, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_37_from107to124.pdf">On perfect numbers which are ratios of two Fibonacci numbers</a>, Annales Mathematicae et Informaticae 37 (2010), pp. 107-124.

%H Bui Minh Phong, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AAPASM_26_from3to8.pdf">Perfect numbers concerning Fibonacci sequence</a>, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 26 (1999), pp. 3-8.

%t Union[Select[First[#]/Last[#]&/@Tuples[Fibonacci[Range[50]],2],IntegerQ]] (* _Harvey P. Dale_, Feb 25 2012 *)

%t mx = 20000; lmt = 40; lst = {}; f[n_] := AppendTo[lst, Select[ Fibonacci@ n/Fibonacci@# & /@ Most@ Divisors@ n, # < mx &]]; Union@ Flatten@ Array[f, lmt] (* _Robert G. Wilson v_, Sep 22 2021 *)

%o (PARI) v=List();for(m=1,100,fordiv(m,d,listput(v,fibonacci(m)/fibonacci(d))));vecsort(Vec(v),,8) \\ _Charles R Greathouse IV_, Jul 26 2011

%o (GAP) Filtered(Set(List(Cartesian([1..21], [1..21]), x -> Fibonacci(x[1] * x[2])/Fibonacci(x[1]))), x -> x < 10000); # _M. Farrokhi D. G._, Jul 24 2020

%Y Cf. A000045, A031122.

%K nonn,easy,nice

%O 1,2

%A _Dan Hoey_

%E More terms from _John W. Layman_, May 22 1999