A106151 In binary representation of n: delete one zero in each contiguous block of zeros.

1, 1, 3, 2, 3, 3, 7, 4, 5, 3, 7, 6, 7, 7, 15, 8, 9, 5, 11, 6, 7, 7, 15, 12, 13, 7, 15, 14, 15, 15, 31, 16, 17, 9, 19, 10, 11, 11, 23, 12, 13, 7, 15, 14, 15, 15, 31, 24, 25, 13, 27, 14, 15, 15, 31, 28, 29, 15, 31, 30, 31, 31, 63, 32, 33, 17, 35, 18, 19, 19, 39, 20, 21, 11, 23, 22, 23

Offset: 1

Author

Reinhard Zumkeller, May 07 2005

Keywords

• nonn
• base

Comments

Equivalently, change bits 10 -> 0. - Michael S. Branicky, Nov 12 2021

Examples

			n=144 = '10010000' -> '101000' = 40 = a(144);
n=145 = '10010001' -> '101001' = 41 = a(145);
n=146 = '10010010' -> '10101'  = 21 = a(146).
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A007088, A030308, A038573, A090078, A175047, A048679, A348710.

Programs

Haskell

import Data.List (group)
a106151 = foldr (\b v -> 2 * v + b) 0 . concatMap
   (\bs'@(b:bs) -> if b == 0 then bs else bs') . group . a030308_row
-- Reinhard Zumkeller, Jul 05 2013

PARI

A106151(n) = if(n<=1, n, if(n%2, 1+(2*A106151((n-1)/2)), A106151(n>>valuation(n, 2))<<(valuation(n, 2)-1))); \\ Antti Karttunen, May 13 2018

PARI

A106151(n) = { my(s=0, i=0); while(n, if(2!=(n%4), s += (n%2)<>= 1); (s); }; \\ Antti Karttunen, Jul 01 2024

Python

def a(n): return int(bin(n).replace("b", "").replace("10", "1"), 2)
print([a(n) for n in range(1, 78)]) # Michael S. Branicky, Nov 12 2021

Formula

a(n) <= n; a(n) = n iff n = 2^k-1: a(A000225(n))=A000225(n);

A000120(a(n)) = A000120(n);

A023416(a(n)) = A023416(n) - A087116(n).

a(n) = b(n, 0), where b(n, r) = if n = 1 then 1 else b(floor(n/2), 1 - n mod 2)*(1 + floor((1 + r + n mod 2)/2)) + n mod 2.

For n <= 1, a(n) = n, and for n > 1, if n is odd, then a(n) = 1+2*a((n-1)/2), otherwise, when n is even, a(n) = (2^(A007814(n)-1)) * a(A000265(n)). - Antti Karttunen, May 13 2018