A031121 Integer ratios of Fibonacci numbers F(m)/F(n).

1, 2, 3, 4, 5, 7, 8, 11, 13, 17, 18, 21, 29, 34, 47, 48, 55, 72, 76, 89, 122, 123, 144, 199, 233, 305, 322, 323, 329, 377, 521, 610, 842, 843, 987, 1292, 1353, 1364, 1597, 2207, 2208, 2255, 2584, 3571, 4181, 5473, 5777, 5778, 5796, 6765, 9349

Offset: 1

Author

Dan Hoey

Keywords

• nonn
• easy
• nice

Comments

Phong shows that no member of this sequence is perfect. - Charles R Greathouse IV, Jul 26 2011

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

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

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• M. Farrokhi D. G., Some remarks on the equation F_n=kF_m in Fibonacci numbers, J. Integer Seq. 10 (2007), no. 5, Article 07.5.7, 9 pp.
• Florian Luca and V. Janitzio Mejía Huguet, On perfect numbers which are ratios of two Fibonacci numbers, Annales Mathematicae et Informaticae 37 (2010), pp. 107-124.
• Bui Minh Phong, Perfect numbers concerning Fibonacci sequence, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 26 (1999), pp. 3-8.

Crossrefs

Cf. A000045, A031122.

Programs

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

Mathematica

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

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

Extensions

More terms from John W. Layman, May 22 1999