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

1,3

Comments

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

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Index entries for sequences related to binary expansion of n

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

Example

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

Prog

(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)<<i; i++); n >>= 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

Crossrefs

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

Sequence in context: A089319 A140396 A117231 * A323467 A341097 A239959

Adjacent sequences: A106148 A106149 A106150 * A106152 A106153 A106154

Keyword

nonn,base

Author

Reinhard Zumkeller, May 07 2005

Status

approved