%I #27 Oct 30 2023 07:48:22
%S 0,1,2,3,4,5,8,9,13,16,21,25,27,32,34,55,64,81,89,125,128,144,169,233,
%T 243,256,377,441,512,610,625,729,987,1024,1156,1597,2048,2187,2197,
%U 2584,3025,3125,4096,4181,6561,6765,7921,8192,9261,10946,15625,16384,17711
%N Powers of Fibonacci numbers.
%C The subset of nontrivial Fibonacci powers [numbers A000045(k)^n which are not in A000045] starts 4, 9, 16, 25, 27, 32, 64, 81, 125, 128, 169, 243, 256, 441, 512, 625, 729, 1024, 1156... - _R. J. Mathar_, Jan 26 2015. These are the initial terms of A254719. - _Reinhard Zumkeller_, Feb 06 2015
%H R. Zumkeller, <a href="/A105317/b105317.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>
%e 2197 = 13^3 = A000045(7)^3, therefore 2197 is a term.
%p N:= 10^6: # to get all terms <= N
%p select(`<=`,{0,1,seq(seq(combinat:-fibonacci(i)^j, i = 3 ..floor(log[phi](sqrt(5)*N^(1/j)+1))),j=1..ilog2(N))},N);
%p # if using Maple 11 or earlier, uncomment the next line
%p # sort(convert(%,list)); # _Robert Israel_, Jan 26 2015
%t lim = 10^5; t = Table[f = Fibonacci[n]; f^Range[Floor[Log[lim]/Log[f]]], {n, 3, Ceiling[Log[GoldenRatio, lim] + 1]}]; Union[{0, 1}, Flatten[t]] (* _T. D. Noe_, Sep 27 2011 *)
%o (Haskell)
%o import Data.Set (singleton, deleteFindMin, insert)
%o a105317 n = a105317_list !! (n-1)
%o a105317_list = 0 : 1 : h 1 (drop 4 a000045_list) (singleton (2, 2)) where
%o h y xs'@(x:xs) s
%o | x < ff = h y xs (insert (x, x) s)
%o | ff == y = h y xs' s'
%o | otherwise = ff : h ff xs' (insert (f * ff, f) s')
%o where ((ff, f), s') = deleteFindMin s
%o -- _Reinhard Zumkeller_, Feb 06 2015
%o (PARI) list(lim)=my(v=List([0]),k=1,f,t); while(k<=lim, listput(v,k); k*=2); k=3; while(k<=lim, listput(v,k); k*=3); k=5; while(k<=lim, listput(v,k); k*=5); k=6; while((f=fibonacci(k++))<=lim, t=1; while((t*=f)<=lim, listput(v,t))); Set(v) \\ _Charles R Greathouse IV_, Oct 03 2016
%Y Subsequences: A056570-A056574, A007598, A000045, A000079, A000244, A000351, A001018, A001022 and A009965.
%Y Cf. A103323, A254719.
%K nonn
%O 1,3
%A _Reinhard Zumkeller_, Apr 25 2005