A090077 -id:A090077 - 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

A090078 In binary expansion of n, reduce contiguous blocks of 0's to 0.

Original entry on oeis.org

0, 1, 2, 3, 2, 5, 6, 7, 2, 5, 10, 11, 6, 13, 14, 15, 2, 5, 10, 11, 10, 21, 22, 23, 6, 13, 26, 27, 14, 29, 30, 31, 2, 5, 10, 11, 10, 21, 22, 23, 10, 21, 42, 43, 22, 45, 46, 47, 6, 13, 26, 27, 26, 53, 54, 55, 14, 29, 58, 59, 30, 61, 62, 63, 2, 5, 10, 11, 10, 21, 22, 23, 10, 21

Offset: 0

Reinhard Zumkeller, Nov 20 2003

			100 -> '1100100' -> 11[00]1[00] -> 11[0]1[0] -> '11010' -> 26=a(100).

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

Cf. A090077, A090079, A007088.

a[n_] := FromDigits[Flatten[Split[IntegerDigits[n, 2]] /. x_List /; x[[1]] == 0 -> {0}], 2]; Array[a, 100, 0] (* Amiram Eldar, Jul 30 2025 *)

a(n)=my(v=binary(n),t); for(i=1,#v, if(v[i], t+=t+1, t%2, t+=t)); t \\ Charles R Greathouse IV, Aug 17 2016

def a(n):
    b = bin(n)[2:]
    while "00" in b: b = b.replace("00", "0")
    return int(b, 2)
print([a(n) for n in range(81)]) # Michael S. Branicky, Jul 27 2022

a(a(n)) = a(n).

a(A090077(n)) = A090077(a(n)) = A090079(n).

A179821 In binary representation of n: replace all blocks of k contiguous ones with binary representation of k.

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 4, 3, 8, 9, 10, 10, 8, 9, 6, 4, 16, 17, 18, 18, 20, 21, 20, 11, 16, 17, 18, 18, 12, 13, 8, 5, 32, 33, 34, 34, 36, 37, 36, 19, 40, 41, 42, 42, 40, 41, 22, 20, 32, 33, 34, 34, 36, 37, 36, 19, 24, 25, 26, 26, 16, 17, 10, 6, 64, 65, 66, 66, 68, 69, 68, 35, 72, 73, 74, 74

Offset: 0

Reinhard Zumkeller, Jul 31 2010

a(n) <= n:
a(A003714(n)) = A003714(n); a(A004780(n)) < A004780(n);
A090077(n) <= a(n).

			n=45->101101->[1]0[11]0[1]->[1]0[2]0[1]->[1]0[10]0[1]->101001->a(45)=41.

Cf. A038374, A069010, A000120, A000225, A007088.

a(2*n) = 2*a(n); a(4*n+1) = 4*a(n)+1.

A348710 In the binary expansion of n, decrease the length of each run of 1-bits by one.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 3, 0, 0, 0, 1, 4, 2, 6, 7, 0, 0, 0, 1, 0, 0, 2, 3, 8, 4, 4, 5, 12, 6, 14, 15, 0, 0, 0, 1, 0, 0, 2, 3, 0, 0, 0, 1, 4, 2, 6, 7, 16, 8, 8, 9, 8, 4, 10, 11, 24, 12, 12, 13, 28, 14, 30, 31, 0, 0, 0, 1, 0, 0, 2, 3, 0, 0, 0, 1, 4, 2, 6, 7, 0, 0, 0

Offset: 0

Kevin Ryde, Oct 30 2021

Equivalently, change bits 01 -> 0, including a 0 reckoned above the most significant 1-bit of n so change there.
A single 1-bit run decreases to nothing.  The Fibbinary numbers (A003714) are those n with only single 1-bits so that a(n) = 0 iff n is in A003714.
a(n) = 1 iff n is in A213540 since those values end with bits 011 (which become 01) and otherwise have only single 1-bits, as do the Fibbinary numbers.
Decreasing each run is the inverse of the increase A175048 so that a(A175048(k)) = k.  This n = A175048(k) is the smallest n with a(n) = k and then other occurrences of k are by inserting single 1-bits into this n, including anywhere above the most significant bit.

			n    = 14551 = binary 111 000 11 0 1 0 111
a(n) =   787 = binary  11 000  1 0   0  11

Cf. A007088 (binary), A175048 (increase 1-bits), A090077 (decrease to single 1-bits).
Cf. A003714 (indices of 0's), A213540 (indices of 1's).
Cf. A106151 (decrease 0-bits), A318921 (decrease each run).

Table[FromDigits[Flatten[Split@IntegerDigits[n,2]/. {1,a___}:>{a}],2],{n,0,82}] (* Giorgos Kalogeropoulos, Nov 01 2021 *)

a(n) = my(v=binary(n),t=0); for(i=2,#v, if(v[i-1]||!v[i], v[t++]=v[i])); fromdigits(v[1..t],2);

def a(n): return int(bin(n).replace("b", "").replace("01", "0"), 2)
print([a(n) for n in range(83)]) # Michael S. Branicky, Oct 31 2021

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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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A090078 In binary expansion of n, reduce contiguous blocks of 0's to 0.

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A179821 In binary representation of n: replace all blocks of k contiguous ones with binary representation of k.

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A348710 In the binary expansion of n, decrease the length of each run of 1-bits by one.

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