A070941 Length of binary representation of 2n+1.

1, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 0

Author

N. J. A. Sloane, May 18 2002

Keywords

• nonn

Comments

Sequence consists of A011782(n) n+1's. - Jon Perry, Apr 04 2004

For n > 0: a(n) = A003314(n+1) - A003314(n) = A123753(n) - A123753(n-1). - Reinhard Zumkeller, Oct 12 2006

For k >= 2, k appears 2^(k-2) times consecutively. - Bernard Schott, Jun 08 2019

Also length of binary representation of 2n. - Michel Marcus, Oct 28 2020

Links

• Index entries for sequences related to binary expansion of n

Crossrefs

Bisection of A070939 and also of A070940.

Programs

Mathematica

Table[IntegerLength[n,2],{n,1,201,2}] (* Harvey P. Dale, May 17 2011 *)

PARI

a(n)=length(binary(2*n+1))

Python

def A070941(n): return n.bit_length()+1 # Chai Wah Wu, Mar 29 2023

Formula

Let b(1)=1, b(n) = a(n-floor(n/2)) + 1, then a(n) = b(n+1). - Benoit Cloitre, Oct 23 2002

G.f.: 1/(1-x) * (1 + Sum_{k>=0} x^2^k). - Ralf Stephan, Apr 15 2002

a(n) = ceiling(log_2(n+1)) + 1 = A029837(n+1) + 1. - Ralf Stephan, Apr 15 2002

a(n) = ceiling(average of previous entries) + 1. - Jon Perry, Apr 04 2004