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URL: https://oeis.org/A105317

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A105317
Powers of Fibonacci numbers.
7
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
1,3
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
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
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Apr 25 2005
STATUS
approved