

A227875


Fibonacci numbers which are perfect powers.


1




OFFSET

1,3


COMMENTS

Also, Fibonacci numbers which are products of Fibonacci numbers (each greater than 1 when the product is greater than 1  see A235383).  Rick L. Shepherd, Feb 19 2014
The terms of the subsequence (1, 8, 144) are the Fibonacci numbers that are powerful numbers.  Robert C. Lyons, Jul 12 2016
Also Fibonacci numbers without any primitive divisors. See [Heuberger & Wagner].  Michel Marcus, Aug 21 2016
It was proved (Bugeaud, Mignotte, and Siksek, 2006, p. 971) that the only perfect powers among the Fibonacci numbers and Lucas numbers are {0, 1, 8, 144} and {1, 4}, respectively.  Daniel Forgues, Apr 09 2018


LINKS

Table of n, a(n) for n=1..4.
Vladica Andrejic, On Fibonacci Powers, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 17 (2006), 3844.
Yann Bugeaud, Florian Luca, Maurice Mignotte, and Samir Siksek, On Fibonacci numbers with few prime divisors, Proc. Japan Acad., 81, Ser. A (2005), pp. 1720.
Yann Bugeaud, Maurice Mignotte, and Samir Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Annals of Mathematics, 163 (2006), pp. 9691018.
Clemens Heuberger, Stephan Wagner, On the monoid generated by a Lucas sequence, arXiv:1606.02639 [math.NT], 2016. Gives the complement sequence w.r.t Fibonacci numbers.
J. Mc Laughlin, Small prime powers in the Fibonacci sequence, arXiv:math/0110150 [math.NT] (2001).
Attila Pethõ, Diophantine properties of linear recursive sequences II, Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 17:2 (2001), pp. 8196.


MATHEMATICA

perfectPowerQ[0] = True; perfectPowerQ[1] = True; perfectPowerQ[n_] := GCD @@ FactorInteger[n][[All, 2]] > 1; Union[Select[Fibonacci /@ Range[0, 20], perfectPowerQ]]


CROSSREFS

Cf. A000045, A072381, A114842.
Sequence in context: A317112 A320396 A112464 * A235383 A275034 A275139
Adjacent sequences: A227872 A227873 A227874 * A227876 A227877 A227878


KEYWORD

nonn,bref,fini,full


AUTHOR

JeanFrançois Alcover, Oct 25 2013


STATUS

approved



