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A105317 Powers of Fibonacci numbers. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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

LINKS

T. D. Noe and 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

EXAMPLE

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

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 *)

PROG

(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

(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

CROSSREFS

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

Cf. A103323, A254719.

Sequence in context: A191851 A063678 A005424 * A094103 A268359 A054181

Adjacent sequences:  A105314 A105315 A105316 * A105318 A105319 A105320

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Apr 25 2005

STATUS

approved

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Last modified February 18 20:32 EST 2018. Contains 299330 sequences. (Running on oeis4.)