A035327 -id:A035327 - cp's OEIS frontend

A003817 a(0) = 0, a(n) = a(n-1) OR n.

0, 1, 3, 3, 7, 7, 7, 7, 15, 15, 15, 15, 15, 15, 15, 15, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63

Offset: 0

Marc LeBrun

Also, 0+1+2+...+n in lunar arithmetic in base 2 written in base 10. - N. J. A. Sloane, Oct 02 2010

For n>0: replace all 0's with 1's in binary representation of n. - Reinhard Zumkeller, Jul 14 2003

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 [From _Reinhard Zumkeller_, Nov 14 2009]
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Reinhard Zumkeller, Logical Convolutions
Index entries for sequences related to dismal (or lunar) arithmetic

This is Guy Steele's sequence GS(6, 6) (see A135416).
Cf. A000004, A142149, A086099, A142150, A142151, A001477, A062383.
Cf. A167832, A167878. - Reinhard Zumkeller, Nov 14 2009
Cf. A179526; subsequence of A007448. - Reinhard Zumkeller, Jul 18 2010
Cf. A265705.

import Data.Bits ((.|.))
a003817 n = if n == 0 then 0 else 2 * a053644 n - 1
a003817_list = scanl (.|.) 0 [1..] :: [Integer]
-- Reinhard Zumkeller, Dec 08 2012, Jan 15 2012

A003817 := n -> n + Bits:-Nand(n, n):
seq(A003817(n), n=0..61); # Peter Luschny, Sep 23 2019

a[0] = 0; a[n_] := a[n] = BitOr[ a[n-1], n]; Table[a[n], {n, 0, 61}] (* Jean-François Alcover, Dec 19 2011 *)
nxt[{n_,a_}]:={n+1,BitOr[a,n+1]}; Transpose[NestList[nxt,{0,0},70]] [[2]] (* Harvey P. Dale, May 06 2016 *)
2^BitLength[Range[0,100]]-1 (* Paolo Xausa, Feb 08 2024 *)

a(n)=1<<(log(2*n+1)\log(2))-1 \\ Charles R Greathouse IV, Dec 08 2011

def a(n): return 0 if n==0 else 1 + 2*a(int(n/2)) # Indranil Ghosh, Apr 28 2017

def A003817(n): return (1<Chai Wah Wu, Jul 17 2024

a(n) = a(n-1) + n*(1-floor(a(n-1)/n)). If 2^(k-1) <= n < 2^k, a(n) = 2^k - 1. - Benoit Cloitre, Aug 25 2002
a(n) = 1 + 2*a(floor(n/2)) for n > 0. - Benoit Cloitre, Apr 04 2003
G.f.: (1/(1-x)) * Sum_{k>=0} 2^k*x^2^k. - Ralf Stephan, Apr 18 2003
a(n) = 2*A053644(n)-1 = A092323(n) + A053644(n). - Reinhard Zumkeller, Feb 15 2004; corrected by Anthony Browne, Jun 26 2016
a(n) = OR{k OR (n-k): 0<=k<=n}. - Reinhard Zumkeller, Jul 15 2008
For n>0: a(n+1) = A035327(n) + n = A035327(n) XOR n. - Reinhard Zumkeller, Nov 14 2009
A092323(n+1) = floor(a(n)/2). - Reinhard Zumkeller, Jul 18 2010
a(n) = A265705(n,0) = A265705(n,n). - Reinhard Zumkeller, Dec 15 2015
a(n) = A062383(n) - 1.
G.f. A(x) satisfies: A(x) = 2*A(x^2)*(1 + x) + x/(1 - x). - Ilya Gutkovskiy, Aug 31 2019
a(n) >= A175039(n) - Austin Shapiro, Dec 29 2022

A233249 a(1)=0; for k >= 1, let prime(k) map to 10...0 with k-1 zeros and let prime(k)*prime(m) map to the concatenation in binary of 2^(k-1) and 2^(m-1). For n >= 2, let the prime power factorization of n be mapped to r(n). a(n) is the term in A114994 which is c-equivalent to r(n) (see there our comment).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 7, 10, 9, 16, 11, 32, 17, 18, 15, 64, 21, 128, 19, 34, 33, 256, 23, 36, 65, 42, 35, 512, 37, 1024, 31, 66, 129, 68, 43, 2048, 257, 130, 39, 4096, 69, 8192, 67, 74, 513, 16384, 47, 136, 73, 258, 131, 32768, 85, 132, 71, 514, 1025, 65536, 75

Offset: 1

Vladimir Shevelev, Dec 06 2013

Let (10...0)_i (i>=0) denote 2^i in binary. Under (10...0)_i^k we understand a concatenation of (10...0)_i k times.
If n=Product_{i=1..m} p_i^t_i is the prime power factorization of n, then in the name r(n)=concatenation{i=1..m} ((10...0_(i-1)^t_i).
Numbers q and s are called c-equivalent if their binary expansions contain the same set of parts of the form 10...0. For example, 14=(1)(1)(10)~(10)(1)(1)=11.
Conversely, if n~n_1 such that n_1 is in A114994 and has c-factorization: n_1 = concatenation{i=m,...,0} ((10...0)i^t_i), one can consider "converse" sequence {s(n)}, where s(n) = Product{i=m..0} p_(i+1)^t_i.
For example, for n=22, n_1=21=((10)^2)(1), and s(22)=3^2*2=18.
The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary binary expansion of k, prepending 0, taking first differences, and reversing again. Then a(n) is the number k such that the k-th composition in standard order consists of the prime indices of n in weakly decreasing order (the partition with Heinz number n). - Gus Wiseman, Apr 02 2020

			n=10=2*5 is mapped to (1)(100)~(100)(1). Since 9 is in A114994, then a(10)=9.
From _Gus Wiseman_, Apr 02 2020: (Start)
The sequence together with the corresponding compositions begins:
   0: ()             128: (8)             2048: (12)
   1: (1)             19: (3,1,1)          257: (8,1)
   2: (2)             34: (4,2)            130: (6,2)
   3: (1,1)           33: (5,1)             39: (3,1,1,1)
   4: (3)            256: (9)             4096: (13)
   5: (2,1)           23: (2,1,1,1)         69: (4,2,1)
   8: (4)             36: (3,3)           8192: (14)
   7: (1,1,1)         65: (6,1)             67: (5,1,1)
  10: (2,2)           42: (2,2,2)           74: (3,2,2)
   9: (3,1)           35: (4,1,1)          513: (9,1)
  16: (5)            512: (10)           16384: (15)
  11: (2,1,1)         37: (3,2,1)           47: (2,1,1,1,1)
  32: (6)           1024: (11)             136: (4,4)
  17: (4,1)           31: (1,1,1,1,1)       73: (3,3,1)
  18: (3,2)           66: (5,2)            258: (7,2)
  15: (1,1,1,1)      129: (7,1)            131: (6,1,1)
  64: (7)             68: (4,3)          32768: (16)
  21: (2,2,1)         43: (2,2,1,1)         85: (2,2,2,1)
For example, the Heinz number of (2,2,1) is 18, and the 21st composition in standard order is (2,2,1), so a(18) = 21.
(End)

Peter J. C. Moses, Table of n, a(n) for n = 1..2500

The sorted version is A114994.
The primorials A002110 map to A246534.
A partial inverse is A333219.
The reversed version is A333220.
Cf. A000120, A029931, A035327, A048793, A066099, A070939, A124767, A124768, A228351, A272020, A333217, A333221.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[2^Accumulate[primeMS[n]]]/2,{n,100}] (* Gus Wiseman, Apr 02 2020 *)

A059893(a(n)) = A333220(n). A124767(a(n)) = A001221(n). - Gus Wiseman, Apr 02 2020

More terms from Peter J. C. Moses, Dec 07 2013

A080079 Least number causing the longest carry sequence when adding numbers <= n to n in binary representation.

Original entry on oeis.org

1, 2, 1, 4, 3, 2, 1, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47

Offset: 1

Reinhard Zumkeller, Jan 26 2003

T(n,k) < T(n,a(n)) = A070940(n) for 1 <= k < a(n) and T(n,k) <= T(n,a(n)) for a(n) <= k <= n, where T is defined as in A080080.

a(n) gives the distance from n to the nearest 2^t > n. - Ctibor O. Zizka, Apr 09 2020

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

Cf. A327489, A004755, A062383, A080080, A240769, A006257.

a080079 n = (length $ takeWhile (< a070940 n) (a080080_row n)) + 1
-- Reinhard Zumkeller, Apr 22 2013

[-n+2*2^Floor(Log(n)/Log(2)): n in [1..80]]; // Vincenzo Librandi, Dec 01 2016

# Alois P. Heinz observes in A327489:
A080079 := n -> 1 + Bits:-Nor(n, n):
# Likewise:
A080079 := n -> 1 + Bits:-Nand(n, n):
seq(A080079(n), n=1..81); # Peter Luschny, Sep 23 2019

Flatten@Table[Nest[Most[RotateRight[#]] &, Range[n], n - 1], {n, 30}] (* Birkas Gyorgy, Feb 07 2011 *)
Table[FromDigits[(IntegerDigits[n, 2] /. {0 -> 1, 1 -> 0}), 2] +
1, {n, 30}] (* Birkas Gyorgy, Feb 07 2011 *)
Table[BitXor[n, 2^IntegerPart[Log[2, n] + 1] - 1] + 1, {n, 30}] (* Birkas Gyorgy, Feb 07 2011 *)
Table[2 2^Floor[Log[2, n]] - n, {n, 30}] (* Birkas Gyorgy, Feb 07 2011 *)
Flatten@Table[Reverse@Range[2^n], {n, 0, 4}] (* Birkas Gyorgy, Feb 07 2011 *)

def A080079(n): return (1 << n.bit_length())-n # Chai Wah Wu, Jun 30 2022

From Benoit Cloitre, Feb 22 2003: (Start)
a(n) = A004755(n) - 2*n.
a(n) = -n + 2*2^floor(log(n)/log(2)). (End)
From Ralf Stephan, Jun 02 2003: (Start)
a(n) = n iff n = 2^k, otherwise a(n) = A035327(n-1).
a(n) = A062383(n) - n. (End)
a(0) = 0, a(2*n) = 2*a(n), a(2*n+1) = 2*a(n)-1 + 2*[n==0]. - Ralf Stephan, Jan 04 2004
a(n) = A240769(n,1); A240769(n, a(n)) = 1. - Reinhard Zumkeller, Apr 13 2014
a(n) = n + 1 - A006257(n). - Reinhard Zumkeller, Apr 14 2014

A037834 a(n) = Sum_{i=1..m} |d(i) - d(i-1)|, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 2, 1, 0, 1, 2, 3, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 2, 1, 0, 1, 2, 3, 2, 3, 4, 3, 2, 3, 4, 5, 4, 3, 4, 3, 2, 1, 2, 3, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 2, 1, 0, 1, 2, 3, 2, 3, 4, 3, 2, 3, 4, 5, 4, 3, 4, 3, 2, 3, 4, 5, 4, 5, 6, 5, 4, 3, 4, 5

Offset: 1

Clark Kimberling

Number of i such that |d(i) - d(i-1)| = 1, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.

a(n)+1 is the number of iterations of the map x -> A035327(x) needed to reach 0 (see A005811(n) for n>=1). - Flávio V. Fernandes, Apr 28 2025

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021.
Index entries for sequences related to binary expansion of n

Cf. A005811, A030308, A035327.

a037834 n = sum $ map fromEnum $ zipWith (/=) (tail bs) bs
            where bs = a030308_row n
-- Reinhard Zumkeller, Feb 20 2014

A037834 := proc(n)
    local dgs ;
    dgs := convert(n,base,2);
    add( abs(op(i,dgs)-op(i-1,dgs)),i=2..nops(dgs)) ;
end proc: # R. J. Mathar, Oct 16 2015

Table[Total@ Flatten@ Map[Abs@ Differences@ # &, Partition[ IntegerDigits[n, 2], 2, 1]], {n, 90}] (* Michael De Vlieger, May 09 2017 *)

def A037834(n): return (n^(n>>1)).bit_count()-1 # Chai Wah Wu, Jul 13 2024

a(n) = A005811(n)-1.

A326703 BII-numbers of chains of nonempty sets.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 16, 17, 24, 32, 34, 40, 64, 65, 66, 68, 69, 70, 72, 80, 81, 88, 96, 98, 104, 128, 256, 257, 384, 512, 514, 640, 1024, 1025, 1026, 1028, 1029, 1030, 1152, 1280, 1281, 1408, 1536, 1538, 1664, 2048, 2056, 2176, 4096, 4097, 4104, 4112, 4113, 4120

Offset: 1

Gus Wiseman, Jul 21 2019

A binary index of n is any position of a 1 in its reversed binary expansion. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, it follows that the BII-number of {{2},{1,3}} is 18.

Elements of a set-system are sometimes called edges. In a chain of sets, every edge is a subset or superset of every other edge.

			The sequence of all chains of nonempty sets together with their BII-numbers begins:
    0: {}
    1: {{1}}
    2: {{2}}
    4: {{1,2}}
    5: {{1},{1,2}}
    6: {{2},{1,2}}
    8: {{3}}
   16: {{1,3}}
   17: {{1},{1,3}}
   24: {{3},{1,3}}
   32: {{2,3}}
   34: {{2},{2,3}}
   40: {{3},{2,3}}
   64: {{1,2,3}}
   65: {{1},{1,2,3}}
   66: {{2},{1,2,3}}
   68: {{1,2},{1,2,3}}
   69: {{1},{1,2},{1,2,3}}
   70: {{2},{1,2},{1,2,3}}
   72: {{3},{1,2,3}}
   80: {{1,3},{1,2,3}}
   81: {{1},{1,3},{1,2,3}}
   88: {{3},{1,3},{1,2,3}}
   96: {{2,3},{1,2,3}}
   98: {{2},{2,3},{1,2,3}}

John Tyler Rascoe, Table of n, a(n) for n = 1..4860

Chains of nonempty sets are counted by A000629.
MM-numbers of chains of multisets are A318991.
BII-numbers of antichains of nonempty sets are A326704.
Cf. A000120, A014466, A029931, A035327, A048793, A070939, A326031, A326701, A326702.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[0,100],stableQ[bpe/@bpe[#],!SubsetQ[#1,#2]&&!SubsetQ[#2,#1]&]&]

from itertools import chain, count, combinations, islice
from sympy.combinatorics.subsets import ksubsets
def subsets(x):
    for i in range(1,len(x)):
        for j in ksubsets(x,i):
            yield(list(j))
def a_gen(): #generator of terms
    yield 0
    for n in count(1):
        t,v,j = [[]],[],0
        for i in chain.from_iterable(combinations(range(1, n+1), r) for r in range(n+1)):
            if n in i:
                t[j].append([list(i)])
        while n:
            t.append([])
            for i in t[j]:
                if len(i[-1]) > 1:
                    for k in list(subsets(i[-1])):
                        t[j+1].append(i.copy()+[k])
            if len(t[j+1]) < 1:
                break
            j += 1
        for j in chain.from_iterable(t):
            v.append(sum(2**(sum(2**(m-1) for m in k)-1) for k in j))
        yield from sorted(v)
A326703_list = list(islice(a_gen(), 55)) # John Tyler Rascoe, Jun 07 2024

A327041 a(n) is the number whose binary indices are the union of the set-system with BII-number n.

Original entry on oeis.org

0, 1, 2, 3, 3, 3, 3, 3, 4, 5, 6, 7, 7, 7, 7, 7, 5, 5, 7, 7, 7, 7, 7, 7, 5, 5, 7, 7, 7, 7, 7, 7, 6, 7, 6, 7, 7, 7, 7, 7, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

Gus Wiseman, Aug 19 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18.

			22 is the BII-number of {{2},{1,2},{1,3}}, and 7 has binary indices {1,2,3}, so a(22) = 7.

Tilman Piesk, Illustration of the first 128 terms

Indices of records are A253317.

Cf. A000120, A001511, A029931, A035327, A048793, A070939, A072639, A326031, A326702, A326947, A261283 (XOR equivalent).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Total[2^Union@@bpe/@bpe[n]]/2,{n,0,100}]

A359194 Binary complement of 3*n.

Original entry on oeis.org

1, 0, 1, 6, 3, 0, 13, 10, 7, 4, 1, 30, 27, 24, 21, 18, 15, 12, 9, 6, 3, 0, 61, 58, 55, 52, 49, 46, 43, 40, 37, 34, 31, 28, 25, 22, 19, 16, 13, 10, 7, 4, 1, 126, 123, 120, 117, 114, 111, 108, 105, 102, 99, 96, 93, 90, 87, 84, 81, 78, 75, 72, 69, 66, 63, 60, 57

Offset: 0

Joshua Searle, Dec 19 2022

The binary complement takes the binary value of a number and turns any 1s to 0s and vice versa. This is equivalent to subtracting from the next larger Mersenne number.
It is currently unknown whether every starting positive integer, upon iteration, reaches 0.
From M. F. Hasler, Dec 26 2022: (Start)
This map enjoys the following properties:
(P1) a(2*n) = a(n)*2 + 1 (since 3*(2*n) is 3*n shifted one binary digit to the left, and the one's complement yields that of 3*n with a '1' appended).
(P2) As an immediate consequence of (P1), all even-indexed values are odd.
(P3) Also from (P1), by immediate induction we have a(2^n) = 2^n-1 for all n >= 0.
(P4) Also from (P1), a(4*n) = a(n)*4 + 3.
(P5) Similarly, a(4*n+1) = a(n)*4 (because the 1's complement of 3 is 0).
(P6) From (P5), a(n) = 0 for all n in A002450 (= (4^k-1)/3). [For the initial value at n = 0 the discrepancy is explained by the fact that the number 0 should be considered to have zero digits, but here the result is computed with 0 considered to have one binary digit.] (End)

			a(7) = 10 because 3*7 = 21 = 10101_2, whose binary complement is 01010_2 = 10.
a(42) = 1 because 3*42 = 126 = 1111110_2, whose binary complement is 0000001_2 = 1.
a(52) = 99 by
  3*n        = binary 10011100
  complement = binary 01100011 = 99.

Michael S. Branicky, Table of n, a(n) for n = 0..10000
Joshua Searle, Collatz-inspired sequences.

Trisection of A035327.
Cf. A002450, A020988 (indices of 1's).
Cf. A256078.

a(n)=if(n, bitneg(3*n, exponent(3*n)+1), 1) \\ Rémy Sigrist, Dec 22 2022

def a(n): return 1 if n == 0 else (m:=3*n)^((1 << m.bit_length()) - 1)
print([a(n) for n in range(67)]) # Michael S. Branicky, Dec 20 2022

a(n) = A035327(3*n).

a(n) = 0 iff n belongs to A002450 \ {0}. - Rémy Sigrist, Dec 22 2022

A005351 Base -2 representation for n regarded as base 2, then evaluated.

Original entry on oeis.org

0, 1, 6, 7, 4, 5, 26, 27, 24, 25, 30, 31, 28, 29, 18, 19, 16, 17, 22, 23, 20, 21, 106, 107, 104, 105, 110, 111, 108, 109, 98, 99, 96, 97, 102, 103, 100, 101, 122, 123, 120, 121, 126, 127, 124, 125, 114, 115, 112, 113, 118, 119, 116, 117, 74, 75, 72, 73, 78, 79, 76

Offset: 0

N. J. A. Sloane

a(n) = n when n is a power of 4. This is because the even-indexed powers of 2 are the same as the even-indexed powers of -2. - Alonso del Arte, Feb 09 2012
a(n) = n if n is a sum of distinct powers of 4. - Michael Somos, Aug 27 2012
Write n = Sum_{i in b(n)} (-2)^(i - 1), which uniquely determines the set of positive integers b(n). Then a(n) = Sum_{i in b(n)} 2^(i - 1). For example, a(7) = 27 because 7 = (-2)^0 + (-2)^1 + (-2)^3 + (-2)^4 and 27 = 2^0 + 2^1 + 2^3 + 2^4. - Gus Wiseman, Jul 26 2019

			2 = 4+(-2)+0 = 110 => 6, 3 = 4+(-2)+1 = 111 => 7, ..., 6 = (16)+(-8)+0+(-2)+0 = 11010 => 26.

M. Gardner, Knotted Doughnuts and Other Mathematical Entertainments. Freeman, NY, 1986, p. 101.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..8191
Joerg Arndt, Matters Computational (The Fxtbook), p. 58-59
Eric Weisstein's World of Mathematics, Negabinary
Wikipedia, Negative base
A. Wilks, Email, May 22 1991

Cf. A039724. Complement of A005352.
Cf. A185269 (primes in this sequence).
Cf. A000120, A029931, A035327, A053985, A065359, A070939, A326032.

a005351 0 = 0
a005351 n = a005351 n' * 2 + m where
   (n', m) = if r < 0 then (q + 1, r + 2) else (q, r)
             where (q, r) = quotRem n (negate 2)
-- Reinhard Zumkeller, Jul 07 2012

a[n_] := Module[{t = 2(4^Floor[ Log[4, Abs[n] + 1] + 2] - 1)/3}, BitXor[n + t, t]]; Table[a[n], {n, 0, 60}] (* Robert G. Wilson v, Jan 24 2005 *)

a(n) = my(t=(32*4^logint(abs(n)+1,4)-2)/3); bitxor(n+t,t); \\ Ruud H.G. van Tol, Oct 18 2023

def A005351(n):
    s, q = '', n
    while q >= 2 or q < 0:
        q, r = divmod(q, -2)
        if r < 0:
            q += 1
            r += 2
        s += str(r)
    return int(str(q)+s[::-1],2) # Chai Wah Wu, Apr 10 2016

a(4n+2) = 4a(n+1)+2, a(4n+3) = 4a(n+1)+3, a(4n+4) = 4a(n+1), a(4n+5) = 4a(n+1)+1, n>-2, a(1)=1. - Ralf Stephan, Apr 06 2004

More terms from Robert G. Wilson v, Jan 24 2005

A256078 Write n in binary, exchange digits '0' <-> '1'.

Original entry on oeis.org

1, 0, 1, 0, 11, 10, 1, 0, 111, 110, 101, 100, 11, 10, 1, 0, 1111, 1110, 1101, 1100, 1011, 1010, 1001, 1000, 111, 110, 101, 100, 11, 10, 1, 0, 11111, 11110, 11101, 11100, 11011, 11010, 11001, 11000, 10111, 10110, 10101, 10100, 10011, 10010, 10001, 10000

Offset: 0

M. F. Hasler, Mar 22 2015

Binary representation of A035327.

A base-2 analog of A048379.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A035327, A048379.

f:= proc(n) local L,i;
  L:= convert(n,base,2);
  add((1-L[i])*10^(i-1),i=1..nops(L))
end proc:
map(f, [$0..100]); # Robert Israel, Sep 17 2024

Table[FromDigits[IntegerDigits[n, 2] /. {0 -> 1, 1 -> 0}], {n, 0, 47}] (* or *)
Table[FromDigits@ IntegerDigits[BitXor[n, 2^IntegerPart[Log[2, n] + 1] - 1], 2], {n, 0, 47}] (* Michael De Vlieger, Mar 22 2015, the latter based on Alonso del Arte at A035327 *)

A256078(n)=!n+eval(Strchr(apply(d->49-d,binary(n))))

def a(n): return int(bin(1 if n==0 else n^((1 << n.bit_length())-1))[2:])
print([a(n) for n in range(48)]) # Michael S. Branicky, Dec 21 2022

A359207 Number of steps to reach 0 starting with n in the map x->A359194(x) (binary complement of 3n), or -1 if 0 is never reached.

Original entry on oeis.org

0, 1, 2, 11, 12, 1, 10, 3, 4, 13, 2, 19, 80, 9, 2, 15, 16, 81, 14, 11, 12, 1, 6, 83, 8, 73, 22, 79, 7572, 5, 18, 75, 76, 7573, 74, 7, 12, 17, 10, 3, 4, 13, 2, 7571, 4, 85, 78, 15, 96, 21, 5498, 91, 72, 13, 6, 7, 56, 13, 82, 3, 20, 5, 98, 15, 16, 21, 14, 7

Offset: 0

Joshua Searle, Dec 20 2022

It is unknown whether each positive starting integer eventually reaches 0.
From Jon E. Schoenfield, Dec 21 2022: (Start)
a(n) == n (mod 4).
a(n) = 1 iff 3*n + 1 = 4^k for some integer k. (End)
All but 10 values under 10^7 have been run to 0. Each of the remaining 10 requires over 2*10^12 steps. They're all in one group that reaches the same high value (nearly 8 million bits wide) after about 2*10^12 steps. The smallest value in this group is 3417582. - Tim Peters, Jun 14 2023

			a(7) = 3 because it takes 3 steps to reach 0: (7, 10, 1, 0).

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Tim Peters, Seeds under 2*10^6 that take more than 10^8 steps to reach 0
Tim Peters, Seeds under 10^7 that take more than 10^9 steps to reach 0
Joshua Searle, Collatz-inspired sequences.

Cf. A035327, A359194, A359208, A359209, A359255.

f[n_] := FromDigits[BitXor[1, IntegerDigits[3*n, 2]], 2]; Array[-1 + Length@ NestWhileList[f, #, # != 0 &] &, 68, 0] (* Michael De Vlieger, Dec 21 2022, faster function by Hans Havermann *)

f(n) = if(n, bitneg(n, exponent(n)+1), 1); \\ A035327
a(n) = my(nb=0, m=n); while (m, m=f(3*m); nb++); nb; \\ Michel Marcus, Dec 21 2022

def f(n): return 1 if n == 0 else (m:=3*n)^((1 << m.bit_length())-1)
def a(n):
    i, fi = 0, n
    while fi != 0: i, fi = i+1, f(fi)
    return i
print([a(n) for n in range(68)]) # Michael S. Branicky, Dec 20 2022

cp's OEIS Frontend

A003817 a(0) = 0, a(n) = a(n-1) OR n.

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A080079 Least number causing the longest carry sequence when adding numbers <= n to n in binary representation.

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A037834 a(n) = Sum_{i=1..m} |d(i) - d(i-1)|, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.

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A326703 BII-numbers of chains of nonempty sets.

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A327041 a(n) is the number whose binary indices are the union of the set-system with BII-number n.

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A359194 Binary complement of 3*n.

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