

A090996


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


8



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
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OFFSET

0,4


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


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n


FORMULA

a(2^k1)=k; a(A004754(k))=1; a(A004758(k))=2.
a(2^k1)=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)n1) for n>0, where w(n) = floor(log_2(n)), that is, A000523(n).  Velin Yanev, Dec 21 2016


EXAMPLE

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)


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


MATHEMATICA

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


PROG

(PARI) a(n) = if(n==0, 0); b=binary(n+1); if(hammingweight(b) == 1, #b1, 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(i1))); r \\ David A. Corneth, Jul 24 2017


CROSSREFS

a(n) = A007814(1+A030101(n)).
Cf. A279209, A279210.
Sequence in context: A283440 A159864 A144790 * A237453 A265754 A089309
Adjacent sequences: A090993 A090994 A090995 * A090997 A090998 A090999


KEYWORD

base,nonn


AUTHOR

Benoit Cloitre, Feb 29 2004


EXTENSIONS

Edited and corrected by Franklin T. AdamsWatters, 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


STATUS

approved



