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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. 68
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Except for the initial term, 2^n appears 2^n times. - Lekraj Beedassy, May 26 2005

a(n) = smallest k such that row k in triangle A265705 contains n. - Reinhard Zumkeller, Dec 17 2015

a(n) = sum of totient function over powers of 2 <= n. - Anthony Browne, Jun 17 2016

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

N. J. A. Sloane, Transforms

R. Stephan, Some divide-and-conquer sequences ...

R. 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) = (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

MAPLE

a:= n-> 2^ilog2(n):

seq(a(n), n=0..80);  # Alois P. Heinz, Dec 20 2016

MATHEMATICA

a[n_] := 2^(Length[ IntegerDigits[n, 2]] - 1); a[0] = 0; Table[a[n], {n, 0, 74}] (* Jean-Fran├žois 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

CROSSREFS

See A000035 for least significant bit(n).

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

From n >= 1 onward, A053644[n] = A062383[n]/2

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.

Sequence in context: A279506 A105678 A028397 * A279170 A039593 A265529

Adjacent sequences:  A053641 A053642 A053643 * A053645 A053646 A053647

KEYWORD

easy,nonn,nice

AUTHOR

Henry Bottomley, Mar 22 2000

STATUS

approved

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Last modified March 24 09:47 EDT 2017. Contains 283985 sequences.