A053644 Most significant bit of n, msb(n); largest power of 2 less than or equal to n; write n in binary and change all but the first digit to zero.

0, 1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64

Offset: 0

Author

Henry Bottomley, Mar 22 2000

Keywords

• nonn
• nice
• easy

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

Crossrefs

See A000035 for least significant bit(n).

MASKTRANS transform of A055975 (prepended with 0), MASKTRANSi transform of A048678.

Bisection of A065267, A065279, A065291, A072376.

First differences of A063915. Cf. A076877, A073121.

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

Cf. A265705, A000010.

Programs

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

Magma

[0] cat [2^Ilog2(n): n in [1..90]]; // Vincenzo Librandi, Dec 11 2018

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 *)

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

Python

def A053644(n): return 1<Chai Wah Wu, Jul 27 2022

Scala

(0 to 127).map(Integer.highestOneBit()) // _Alonso del Arte, Feb 26 2020

Formula

a(n) = a(floor(n / 2)) * 2.

a(n) = 2^A000523(n).

From n >= 1 onward, A053644(n) = A062383(n)/2.

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) = (A003817(n) + 1)/2 = A091940(n) + 1. - Reinhard Zumkeller, Feb 15 2004

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