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 A227875 Fibonacci numbers which are perfect powers. 2
 0, 1, 8, 144 (list; graph; refs; listen; history; text; internal format)
 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 Vladica Andrejic, On Fibonacci Powers, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 17 (2006), 38-44. Yann Bugeaud, Florian Luca, Maurice Mignotte, and Samir Siksek, On Fibonacci numbers with few prime divisors, Proc. Japan Acad., 81, Ser. A (2005), pp. 17-20. 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. 969-1018. 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. 81-96. 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 Jean-François Alcover, Oct 25 2013 STATUS approved

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Last modified October 15 20:04 EDT 2019. Contains 328037 sequences. (Running on oeis4.)