A053645 Distance to largest power of 2 less than or equal to n; write n in binary, change the first digit to zero, and convert back to decimal.

0, 0, 1, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

Offset: 1

Author

Henry Bottomley, Mar 22 2000

Keywords

• nonn
• base
• easy

Comments

Triangle read by rows in which row n lists the first 2^n nonnegative integers (A001477), n >= 0. Right border gives A000225. Row sums give A006516. See example. - Omar E. Pol, Oct 17 2013

Without the initial zero also: zeroless numbers in base 3 (A032924: 1, 2, 11, 12, 21, ...), ternary digits decreased by 1 and read as binary. - M. F. Hasler, Jun 22 2020

Examples

			From _Omar E. Pol_, Oct 17 2013: (Start)
Written as an irregular triangle the sequence begins:
  0;
  0,1;
  0,1,2,3;
  0,1,2,3,4,5,6,7;
  0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15;
  ...
(End)
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• J.-P. Allouche and J. Shallit, The ring of k-regular sequences, preprint, Theoretical Computer Sci., 98 (1992), 163-197.
• J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197 (see Ex. 24).
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000225, A000523, A002262, A004760, A006257, A006516, A030308, A036987, A053644, A062050, A083741, A160588.

Programs

Haskell

a053645 1 = 0
a053645 n = 2 * a053645 n' + b  where (n', b) = divMod n 2
-- Reinhard Zumkeller, Aug 28 2014
a053645_list = concatMap (0 `enumFromTo`) a000225_list
-- Reinhard Zumkeller, Feb 04 2013, Mar 23 2012

Magma

[n - 2^Ilog2(n): n in [1..70]]; // Vincenzo Librandi, Jul 18 2019

Maple

seq(n - 2^ilog2(n), n=1..1000); # Robert Israel, Dec 23 2015

Mathematica

Table[n - 2^Floor[Log2[n]], {n, 100}] (* IWABUCHI Yu(u)ki, May 25 2017 *)
Table[FromDigits[Rest[IntegerDigits[n, 2]], 2], {n, 100}] (* IWABUCHI Yu(u)ki, May 25 2017 *)

PARI

a(n)=n-2^(#binary(n)-1) \\ Charles R Greathouse IV, Sep 02 2015

Python

def a(n): return n - 2**(n.bit_length()-1)
print([a(n) for n in range(1, 85)]) # Michael S. Branicky, Jul 03 2021

Python

def A053645(n): return n&(1<Chai Wah Wu, Jan 22 2023

Formula

a(n) = n - 2^A000523(n).

G.f.: 1/(1-x) * ((2x-1)/(1-x) + Sum_{k>=1} 2^(k-1)*x^2^k). - Ralf Stephan, Apr 18 2003

a(n) = (A006257(n)-1)/2. - N. J. A. Sloane, May 16 2003

a(1) = 0, a(2n) = 2a(n), a(2n+1) = 2a(n) + 1. - N. J. A. Sloane, Sep 13 2003

a(n) = A062050(n) - 1. - N. J. A. Sloane, Jun 12 2004

a(A004760(n+1)) = n. - Reinhard Zumkeller, May 20 2009

a(n) = f(n-1,1) with f(n,m) = if n < m then n else f(n-m,2*m). - Reinhard Zumkeller, May 20 2009

Conjecture: a(n) = (1 - A036987(n-1))*(1 + a(n-1)) for n > 1 with a(1) = 0. - Mikhail Kurkov, Jul 16 2019