OFFSET
1,2
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.
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from John W. Layman, May 22 1999
STATUS
approved