A090078 -id:A090078 - cp's OEIS frontend

A090079 In binary expansion of n: reduce contiguous blocks of 0's to 0 and contiguous blocks of 1's to 1.

0, 1, 2, 1, 2, 5, 2, 1, 2, 5, 10, 5, 2, 5, 2, 1, 2, 5, 10, 5, 10, 21, 10, 5, 2, 5, 10, 5, 2, 5, 2, 1, 2, 5, 10, 5, 10, 21, 10, 5, 10, 21, 42, 21, 10, 21, 10, 5, 2, 5, 10, 5, 10, 21, 10, 5, 2, 5, 10, 5, 2, 5, 2, 1, 2, 5, 10, 5, 10, 21, 10, 5, 10, 21, 42, 21, 10, 21, 10, 5, 10, 21, 42, 21

Offset: 0

Reinhard Zumkeller, Nov 20 2003

a(a(n))=a(n); a(n)=A090078(A090077(n))=A090077(A090078(n)).

All terms are without consecutive equal binary digits: a(A000975(n)) = A000975(n) and a(m) <> A000975(n) for m < A000975(n). - Reinhard Zumkeller, Feb 16 2013

			100 -> '1100100' -> [11][00][1][00] -> [1][0][1][0] -> '1010' ->
10=a(100).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n

Cf. A007088, A090077, A090078, A090080.

Cf. A030308, A005811.

a090079 = foldr (\b v -> 2 * v + b) 0 . map head . group . a030308_row
-- Reinhard Zumkeller, Feb 16 2013

Table[FromDigits[#, 2] &@ Map[First, Split@ IntegerDigits[n, 2]], {n, 0, 83}] (* Michael De Vlieger, Dec 12 2016 *)
FromDigits[Split[IntegerDigits[#,2]][[All,1]],2]&/@Range[0,90] (* Harvey P. Dale, Oct 10 2017 *)

from itertools import groupby
def a(n): return int("".join(k for k, g in groupby(bin(n)[2:])), 2)
print([a(n) for n in range(84)]) # Michael S. Branicky, Jul 23 2022

Conjecture: a(n) = (2^(A005811(n)+1) + (1-(-1)^n)/2 - 2)/3. - Velin Yanev, Dec 12 2016

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

Original entry on oeis.org

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

Reinhard Zumkeller, May 07 2005

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

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

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

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

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

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

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

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

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

A090077 In binary expansion of n: reduce contiguous blocks of 1's to 1.

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 2, 1, 8, 9, 10, 5, 4, 5, 2, 1, 16, 17, 18, 9, 20, 21, 10, 5, 8, 9, 10, 5, 4, 5, 2, 1, 32, 33, 34, 17, 36, 37, 18, 9, 40, 41, 42, 21, 20, 21, 10, 5, 16, 17, 18, 9, 20, 21, 10, 5, 8, 9, 10, 5, 4, 5, 2, 1, 64, 65, 66, 33, 68, 69, 34, 17, 72, 73, 74, 37, 36, 37, 18, 9

Offset: 0

Reinhard Zumkeller, Nov 20 2003

			100 -> '1100100' -> [11]00[1]00 -> [1]00[1]00 -> '100100' -> 36=a(100).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n

Cf. A007088, A090079, A090078, A348710.

Array[FromDigits[Flatten[Split@ IntegerDigits[#, 2] /. w_List /; First[w] == 1 -> {1}], 2] &, 80, 0] (* Michael De Vlieger, Jul 28 2022 *)

def a(n):
    b = bin(n)[2:]
    while "11" in b: b = b.replace("11", "1")
    return int(b, 2)
print([a(n) for n in range(81)]) # Michael S. Branicky, Jul 27 2022

a(a(n)) = a(n); a(A090078(n)) = A090078(a(n)) = A090079(n).

a(A003714(n)) = A003714(n); a(A004780(n)) < A004780(n); a(n) <= A179821(n); A085357(a(n)) = 1. - Reinhard Zumkeller, Jul 31 2010

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A090079 In binary expansion of n: reduce contiguous blocks of 0's to 0 and contiguous blocks of 1's to 1.

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

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A090077 In binary expansion of n: reduce contiguous blocks of 1's to 1.

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