OFFSET
0,3
COMMENTS
Except for the initial term, 2^n appears 2^n times. - Lekraj Beedassy, May 26 2005
a(n) is the smallest k such that row k in triangle A265705 contains n. - Reinhard Zumkeller, Dec 17 2015
a(n) is the sum of totient function over powers of 2 <= n. - Anthony Browne, Jun 17 2016
Given positive n, reverse the bits of n and divide by 2^floor(log_2 n). Numerators are in A030101. Ignoring the initial 0, denominators are in this sequence. - Alonso del Arte, Feb 11 2020
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
FORMULA
a(n) = a(floor(n / 2)) * 2.
a(n) = 2^A000523(n).
a(0) = 0, a(1) = 1 and a(n+1) = a(n)*floor(n/a(n)). - Benoit Cloitre, Aug 17 2002
G.f.: 1/(1 - x) * (x + Sum_{k >= 1} 2^(k - 1)*x^2^k). - Ralf Stephan, Apr 18 2003
a(n) = Sum_{k = 1..n} (floor(2^k/k) - floor((2^k - 1)/k))*A000010(k). - Anthony Browne, Jun 17 2016
a(2^m+k) = 2^m, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Aug 07 2016
MAPLE
a:= n-> 2^ilog2(n):
seq(a(n), n=0..80); # Alois P. Heinz, Dec 20 2016
MATHEMATICA
A053644[n_] := 2^(Length[ IntegerDigits[n, 2]] - 1); A053644[0] = 0; Table[A053644[n], {n, 0, 74}] (* Jean-François Alcover, Dec 01 2011 *)
nv[n_] := Module[{c = 2^n}, Table[c, {c}]]; Join[{0}, Flatten[Array[nv, 7, 0]]] (* Harvey P. Dale, Jul 17 2012 *)
PROG
(Haskell)
a053644 n = if n <= 1 then n else 2 * a053644 (div n 2)
-- Reinhard Zumkeller, Aug 28 2014
a053644_list = 0 : concat (iterate (\zs -> map (* 2) (zs ++ zs)) [1])
-- Reinhard Zumkeller, Dec 08 2012, Oct 21 2011, Oct 17 2010
(PARI) a(n)=my(k=1); while(k<=n, k<<=1); k>>1 \\ Charles R Greathouse IV, May 27 2011
(PARI) a(n) = if(!n, 0, 2^exponent(n)) \\ Iain Fox, Dec 10 2018
(Python)
def a(n): return 0 if n==0 else 2**(len(bin(n)[2:]) - 1) # Indranil Ghosh, May 25 2017
(Magma) [0] cat [2^Ilog2(n): n in [1..90]]; // Vincenzo Librandi, Dec 11 2018
(Scala) (0 to 127).map(Integer.highestOneBit(_)) // Alonso del Arte, Feb 26 2020
(Python)
def A053644(n): return 1<<n.bit_length()-1 if n else 0 # Chai Wah Wu, Jul 27 2022
CROSSREFS
See A000035 for least significant bit(n).
This is Guy Steele's sequence GS(5, 5) (see A135416).
Equals for n >= 1 the first right hand column of A160464. - Johannes W. Meijer, May 24 2009
Diagonal of A088370. - Alois P. Heinz, Oct 28 2011
KEYWORD
nonn,nice,easy
AUTHOR
Henry Bottomley, Mar 22 2000
STATUS
approved