A090996 Number of leading 1's in binary expansion of n.

0, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 2, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2

Offset: 0

Author

Benoit Cloitre, Feb 29 2004

Keywords

• nonn
• base

Comments

Mirror of triangle A065120. See example. - Omar E. Pol, Oct 17 2013

a(n) is also the least part in the integer partition having viabin number n. The viabin number of an integer partition is defined in the following way. Consider the southeast border of the Ferrers board of the integer partition and consider the binary number obtained by replacing each east step with 1 and each north step, except the last one, with 0. The corresponding decimal form is, by definition, the viabin number of the given integer partition. "Viabin" is coined from "via binary". For example, consider the integer partition [2,2,2,1]. The southeast border of its Ferrers board yields 10100, leading to the viabin number 20. - Emeric Deutsch, Jul 24 2017

Examples

			In binary : 14=1110 and there are 3 leading 1's, so a(14)=3.
From _Omar E. Pol_, Oct 17 2013: (Start)
Written as an irregular triangle with row lengths A011782 the sequence begins:
0;
1;
1,2;
1,1,2,3;
1,1,1,1,2,2,3,4;
1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5;
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,5,6;
Right border gives A001477. Row sums give A000225.
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..65535 (first 10001 terms from Vincenzo Librandi)
• Index entries for sequences related to binary expansion of n

Crossrefs

a(n) = A007814(1+A030101(n)).

Cf. A279209, A279210, A360189.

Programs

Maple

a := proc(n) if type(log[2](n+1), integer) then log[2](n+1) else a(floor((1/2)*n)) end if end proc: seq(a(n), n = 0 .. 200); # Emeric Deutsch, Jul 24 2017
# second Maple program:
b:= proc(n, t) `if`(n=0, t,
      b(iquo(n, 2, 'm'), m*(t+1)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..127);  # Alois P. Heinz, Mar 06 2023

Mathematica

Join[{0},Table[Length@First@Split@IntegerDigits[n,2],{n,30}]] (* Birkas Gyorgy, Mar 09 2011 *) (* adapted by Vincenzo Librandi, Dec 23 2016 *)

PARI

a(n) = if(n==0, 0); b=binary(n+1); if(hammingweight(b) == 1, #b-1, a(n\2)) \\ David A. Corneth, Jul 24 2017

PARI

a(n) = if(n==0, 0); my(b = binary(n), r = #b); for(i=2, #b, if(!b[i], return(i-1))); r \\ David A. Corneth, Jul 24 2017

Formula

a(2^k-1)=k; a(A004754(k))=1; a(A004758(k))=2.

a(2^k-1)=k; for any other n, a(n) = a(floor(n/2)).

a(n) = f(n, 0) with f(n, x) = if n < 2 then n + x else f([n/2], (x+1)*(n mod 2)). - Reinhard Zumkeller, Feb 02 2007

Conjecture: a(n) = w(n+1)*(w(n+1)-w(n)+1) - w(2^(w(n+1)+1)-n-1) for n>0, where w(n) = floor(log_2(n)), that is, A000523(n). - Velin Yanev, Dec 21 2016

a(n) = A360189(n-1,floor(log_2(n))). - Alois P. Heinz, Mar 06 2023

Extensions

Edited and corrected by Franklin T. Adams-Watters, Apr 08 2006

Sequence had accidentally been shifted left by one step, which was corrected and term a(0)=0 added by Antti Karttunen, Jan 01 2007