A089309 Write n in binary; a(n) = length of the rightmost run of 1's.

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

Offset: 0

Author

N. J. A. Sloane, Dec 22 2003

Keywords

• nonn
• base

Comments

Equivalent to: remove trailing zeros, add one, count trailing zeros. - Ralf Stephan, Aug 31 2013

a(n) is also the difference between the two largest distinct parts in the integer partition having viabin number n (we assume that 0 is a part). 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. Note that a(20) = 1 = the difference between the two largest distinct parts of the partition [2,2,2,1]. - Emeric Deutsch, Aug 17 2017

Examples

			13 = 1101 so a(13) = 1.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..16384
• Francis Laclé, 2-adic parity explorations of the 3n+ 1 problem, hal-03201180v2 [cs.DM], 2021.
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A089310, A089311, A089312, A089313.

Programs

Maple

a := proc(n) if n = 0 then 0 elif `mod`(n, 2) = 0 then a((1/2)*n) elif `mod`(n, 4) = 1 then 1 else 1+a((1/2)*n-1/2) end if end proc: seq(a(n), n = 0 .. 104); # Emeric Deutsch, Aug 17 2017

Mathematica

Table[If[n == 0, 0, Length@ Last@ Select[Split@ IntegerDigits[n, 2], First@ # == 1 &]], {n, 0, 104}] (* Michael De Vlieger, Aug 17 2017 *)

PARI

a(n) = if (n==0, 0, valuation(n/2^valuation(n, 2)+1, 2)); \\ Ralf Stephan, Aug 31 2013; Michel Marcus, Apr 30 2020

Python

def A089309(n): return (~((m:=n>>(~n&n-1).bit_length())+1)&m).bit_length() # Chai Wah Wu, Jul 13 2022

Formula

a(2*n) = a(n), a(2*n+1) = A007814(2*n+2) = A001511(n+1). - Ralf Stephan, Jan 31 2004

a(0) = 0, a(2*n) = a(n), a(4*n+1) = 1, a(4*n+3) = 1 + a(2*n+1) (the Maple program makes use of these equations). - Emeric Deutsch, Aug 17 2017

Extensions

More terms from Vladeta Jovovic and John W. Layman, Jan 21 2004