A105317 Powers of Fibonacci numbers.

0, 1, 2, 3, 4, 5, 8, 9, 13, 16, 21, 25, 27, 32, 34, 55, 64, 81, 89, 125, 128, 144, 169, 233, 243, 256, 377, 441, 512, 610, 625, 729, 987, 1024, 1156, 1597, 2048, 2187, 2197, 2584, 3025, 3125, 4096, 4181, 6561, 6765, 7921, 8192, 9261, 10946, 15625, 16384, 17711

Offset: 1

Author

Reinhard Zumkeller, Apr 25 2005

Keywords

• nonn

Comments

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

Examples

			2197 = 13^3 = A000045(7)^3, therefore 2197 is a term.
		

Links

• R. Zumkeller, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• Eric Weisstein's World of Mathematics, Fibonacci Number

Crossrefs

Subsequences: A056570-A056574, A007598, A000045, A000079, A000244, A000351, A001018, A001022 and A009965.

Cf. A103323, A254719.

Programs

Haskell

import Data.Set (singleton, deleteFindMin, insert)
a105317 n = a105317_list !! (n-1)
a105317_list = 0 : 1 : h 1 (drop 4 a000045_list) (singleton (2, 2)) where
  h y xs'@(x:xs) s
    | x < ff    = h y xs (insert (x, x) s)
    | ff == y   = h y xs' s'
    | otherwise = ff : h ff xs' (insert (f * ff, f) s')
    where ((ff, f), s') = deleteFindMin s
-- Reinhard Zumkeller, Feb 06 2015

Maple

N:= 10^6: # to get all terms <= N
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);
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(%,list)); # Robert Israel, Jan 26 2015

Mathematica

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 *)

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