A116361 -id:A116361 - cp's OEIS frontend

A116362 Smallest m such that A116361(m) = n.

0, 1, 3, 7, 11, 23, 39, 79, 143, 287, 543, 1087, 2111, 4223, 8319, 16639, 33023, 66047, 131583, 263167, 525311, 1050623, 2099199, 4198399, 8392703, 16785407, 33562623, 67125247, 134234111, 268468223, 536903679, 1073807359, 2147549183, 4295098367

Offset: 0

Reinhard Zumkeller, Feb 04 2006

a(n) = 2*a(n-1) + 1 - 0^(n mod 2) * 2^floor(n/2) for n>2.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-6,4).

Cf. A116361.

concat(0, Vec(-x*(4*x^4-4*x^3-2*x^2+1)/((x-1)*(2*x-1)*(2*x^2-1)) + O(x^100))) \\ Colin Barker, Feb 09 2015

a(n) = 2^(n-1) + 2^floor((n+1)/2) - 1 for n > 1.

a(n) = 3*a(n-1)-6*a(n-3)+4*a(n-4) for n>5. G.f.: -x*(4*x^4-4*x^3-2*x^2+1) / ((x-1)*(2*x-1)*(2*x^2-1)). - Colin Barker, Mar 29 2013 and Feb 09 2015

A003714 Fibbinary numbers: if n = F(i1) + F(i2) + ... + F(ik) is the Zeckendorf representation of n (i.e., write n in Fibonacci number system) then a(n) = 2^(i1 - 2) + 2^(i2 - 2) + ... + 2^(ik - 2). Also numbers whose binary representation contains no two adjacent 1's.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 10, 16, 17, 18, 20, 21, 32, 33, 34, 36, 37, 40, 41, 42, 64, 65, 66, 68, 69, 72, 73, 74, 80, 81, 82, 84, 85, 128, 129, 130, 132, 133, 136, 137, 138, 144, 145, 146, 148, 149, 160, 161, 162, 164, 165, 168, 169, 170, 256, 257, 258, 260, 261, 264

Offset: 0

N. J. A. Sloane

The name "Fibbinary" is due to Marc LeBrun.
"... integers whose binary representation contains no consecutive ones and noticed that the number of such numbers with n bits was fibonacci(n)". [posting to sci.math by Bob Jenkins (bob_jenkins(AT)burtleburtle.net), Jul 17 2002]
From Benoit Cloitre, Mar 08 2003: (Start)
A number m is in the sequence if and only if C(3m, m) (or equally, C(3m, 2m)) is odd.
a(n) == A003849(n) (mod 2). (End)
Numbers m such that m XOR 2*m = 3*m. - Reinhard Zumkeller, May 03 2005. [This implies that A003188(2*a(n)) = 3*a(n) holds for all n.]
Numbers whose base-2 representation contains no two adjacent ones. For example, m = 17 = 10001_2 belongs to the sequence, but m = 19 = 10011_2 does not. - Ctibor O. Zizka, May 13 2008
m is in the sequence if and only if the central Stirling number of the second kind S(2*m, m) = A007820(m) is odd. - O-Yeat Chan (math(AT)oyeat.com), Sep 03 2009
A000120(3*a(n)) = 2*A000120(a(n)); A002450 is a subsequence.
Every nonnegative integer can be expressed as the sum of two terms of this sequence. - Franklin T. Adams-Watters, Jun 11 2011
Subsequence of A213526. - Arkadiusz Wesolowski, Jun 20 2012
This is also the union of A215024 and A215025 - see the Comment in A014417. - N. J. A. Sloane, Aug 10 2012
The binary representation of each term m contains no two adjacent 1's, so we have (m XOR 2m XOR 3m) = 0, and thus a two-player Nim game with three heaps of (m, 2m, 3m) stones is a losing configuration for the first player. - V. Raman, Sep 17 2012
Positions of zeros in A014081. - John Keith, Mar 07 2022
These numbers are similar to Fibternary numbers A003726, Tribbinary numbers A060140 and Tribternary numbers. This sequence is a subsequence of Fibternary numbers A003726. The number of Fibbinary numbers less than any power of two is a Fibonacci number. We can generate this sequence recursively: start with 0 and 1; then, if x is in the sequence add 2x and 4x+1 to the sequence. The Fibbinary numbers have the property that the n-th Fibbinary number is even if the n-th term of the Fibonacci word is a. Respectively, the n-th Fibbinary number is odd (of the form 4x+1) if the n-th term of the Fibonacci word is b. Every number has a Fibbinary multiple. - Tanya Khovanova and PRIMES STEP Senior, Aug 30 2022
This is the ordered set S of numbers defined recursively by: 0 is in S; if x is in S, then 2*x and 4*x + 1 are in S. See Kimberling (2006) Example 3, in references below. - Harry Richman, Jan 31 2024

			From _Joerg Arndt_, Jun 11 2011: (Start)
In the following, dots are used for zeros in the binary representation:
  a(n)  binary(a(n))  n
    0:    .......     0
    1:    ......1     1
    2:    .....1.     2
    4:    ....1..     3
    5:    ....1.1     4
    8:    ...1...     5
    9:    ...1..1     6
   10:    ...1.1.     7
   16:    ..1....     8
   17:    ..1...1     9
   18:    ..1..1.    10
   20:    ..1.1..    11
   21:    ..1.1.1    12
   32:    .1.....    13
   33:    .1....1    14
   34:    .1...1.    15
   36:    .1..1..    16
   37:    .1..1.1    17
   40:    .1.1...    18
   41:    .1.1..1    19
   42:    .1.1.1.    20
   64:    1......    21
   65:    1.....1    22
(End)

Donald E. Knuth, The Art of Computer Programming: Fundamental Algorithms, Vol. 1, 2nd ed., Addison-Wesley, 1973, pp. 85, 493.

G. C. Greubel and T. D. Noe, Table of n, a(n) for n = 0..5000 (terms 0 to 1363 by T. D. Noe)
J.-P. Allouche, J. Shallit, and G. Skordev, Self-generating sets, integers with missing blocks and substitutions, Discrete Math., Vol. 292, No. 1-3 (2005), pp. 1-15.
Joerg Arndt, Matters Computational (The Fxtbook), pp. 74-77, pp. 754-756.
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Marc Chamberland and Karl Dilcher, A Binomial Sum Related to Wolstenholme's Theorem, J. Number Theory, Vol. 171, Issue 11 (Nov. 2009), pp. 2659-2672. See Lemma 4.2 p. 2668.
O-Yeat Chan and Dante Manna, Divisibility properties of Stirling numbers of the second kind.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
David Eppstein, Generating fibbinary numbers, three ways, 2021.
Clark Kimberling, Affinely recursive sets and orderings of languages, Discrete Math., Vol. 274, No. 1-3 (2004), pp. 147-160.
Roman S. Klyujkov, Quick Fibbinary Numbers Addition, C++ function with test program.
Ron Knott, Rabbit Sequence in Zeckendorf Expansion (A003714).
Linus Lindroos, Andrew Sills, and Hua Wang, Odd fibbinary numbers and the golden ratio, Fib. Q., Vol. 52, No. 1 (2014), pp. 61-65; alternative link.
A. J. Macfarlane, On the fibbinary numbers and the Wythoff array, arXiv:2405.18128 [math.CO], 2024. See pages 1, 5.
Douglas M. McKenna, Fibbinary Zippers in a Monoid of Toroidal Hamiltonian Cycles that Generate Hilbert-Style Square-Filling Curves, ECA 2:2 (2022) Article S2R13.
Joris Nieuwveld, Fractions, Functions and Folding. A Novel Link between Continued Fractions, Mahler Functions and Paper Folding, Master's Thesis, arXiv:2108.11382 [math.NT], 2021.
Yaron Shany and Amit Berman, The Generating Idempotent Is a Minimum-Weight Codeword for Some Binary BCH Codes, arXiv:2408.08218 [cs.IT], 2024. See p. 10.
N. J. A. Sloane, Table of n, a(n) (base 10), a(n) (base 2), for n = 0..1000.
Chai Wah Wu, Record values in appending and prepending bitstrings to runs of binary digits, arXiv:1810.02293 [math.NT], 2018.
Index entries for 2-automatic sequences.
Index entries for sequences defined by congruent products between domains N and GF(2)[X].
Index entries for sequences defined by congruent products under XOR.

A007088(a(n)) = A014417(n) (same sequence in binary). Complement: A004780. Char. function: A085357. Even terms: A022340, odd terms: A022341. First difference: A129761.
Other sequences based on similar restrictions on binary expansion: A003726 & A278038, A003754, A048715, A048718, A107907, A107909.
Cf. A000045, A005203, A005590, A007895, A037011, A048728, A048679, A056017, A060112, A072649, A083368, A089939, A106027, A106028, A116361.
3*a(n) is in A001969.
Cf. also A215022-A215025, A242407.
Cf. A014081 (count 11 bits).

import Data.Set (Set, singleton, insert, deleteFindMin)
a003714 n = a003714_list !! n
a003714_list = 0 : f (singleton 1) where
   f :: Set Integer -> [Integer]
   f s = m : (f $ insert (4*m + 1) $ insert (2*m) s')
         where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 03 2012, Feb 07 2012

A003714 := proc(n)
    option remember;
    if n < 3 then
        n ;
    else
        2^(A072649(n)-1) + procname(n-combinat[fibonacci](1+A072649(n))) ;
    end if;
end proc:
seq(A003714(n),n=0..10) ;
# To produce a table giving n, a(n) (base 10), a(n) (base 2) - from N. J. A. Sloane, Sep 30 2018
# binary: binary representation of n, in human order
binary:=proc(n) local t1,L;
if n<0 then ERROR("n must be nonnegative"); fi;
if n=0 then return([0]); fi;
t1:=convert(n,base,2); L:=nops(t1);
[seq(t1[L+1-i],i=1..L)];
end;
for n from 0 to 100 do t1:=A003714(n); lprint(n, t1, binary(t1)); od:

fibBin[n_Integer] := Block[{k = Ceiling[Log[GoldenRatio, n Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; FromDigits[fr, 2]]; Table[fibBin[n], {n, 0, 61}] (* Robert G. Wilson v, Sep 18 2004 *)
Select[Range[0, 270], ! MemberQ[Partition[IntegerDigits[#, 2], 2, 1], {1, 1}] &] (* Harvey P. Dale, Jul 17 2011 *)
Select[Range[256], BitAnd[#, 2 #] == 0 &] (* Alonso del Arte, Jun 18 2012 *)
With[{r = Range[10^5]}, Pick[r, BitAnd[r, 2 r], 0]] (* Eric W. Weisstein, Aug 18 2017 *)
Select[Range[0, 299], SequenceCount[IntegerDigits[#, 2], {1, 1}] == 0 &] (* Requires Mathematica version 10 or later. -- Harvey P. Dale, Dec 06 2018 *)

msb(n)=my(k=1); while(k<=n, k<<=1); k>>1
for(n=1,1e4,k=bitand(n,n<<1);if(k,n=bitor(n,msb(k)-1),print1(n", "))) \\ Charles R Greathouse IV, Jun 15 2011

select( is_A003714(n)=!bitand(n,n>>1), [0..266])
{(next_A003714(n,t)=while(t=bitand(n+=1,n<<1), n=bitor(n,1<A003714(t)) \\ M. F. Hasler, Nov 30 2021

for n in range(300):
    if 2*n & n == 0:
        print(n, end=",") # Alex Ratushnyak, Jun 21 2012

def A003714(n):
    tlist, s = [1,2], 0
    while tlist[-1]+tlist[-2] <= n:
        tlist.append(tlist[-1]+tlist[-2])
    for d in tlist[::-1]:
        s *= 2
        if d <= n:
            s += 1
            n -= d
    return s # Chai Wah Wu, Jun 14 2018

def fibbinary():
    x = 0
    while True:
        yield x
        y = ~(x >> 1)
        x = (x - y) & y # Falk Hüffner, Oct 23 2021
(C++)
/* start with x=0, then repeatedly call x=next_fibrep(x): */
ulong next_fibrep(ulong x)
{
    // 2 examples:         //  ex. 1             //  ex.2
    //                     // x == [*]0 010101   // x == [*]0 01010
    ulong y = x | (x>>1);  // y == [*]? 011111   // y == [*]? 01111
    ulong z = y + 1;       // z == [*]? 100000   // z == [*]? 10000
    z = z & -z;            // z == [0]0 100000   // z == [0]0 10000
    x ^= z;                // x == [*]0 110101   // x == [*]0 11010
    x &= ~(z-1);           // x == [*]0 100000   // x == [*]0 10000
    return x;
}
/* Joerg Arndt, Jun 22 2012 */

(0 to 255).filter(n => (n & 2 * n) == 0) // Alonso del Arte, Apr 12 2020
(C#)
public static bool IsFibbinaryNum(this int n) => ((n & (n >> 1)) == 0) ? true : false; // Frank Hollstein, Jul 07 2021

No two adjacent 1's in binary expansion.
Let f(x) := Sum_{n >= 0} x^Fibbinary(n). (This is the generating function of the characteristic function of this sequence.) Then f satisfies the functional equation f(x) = x*f(x^4) + f(x^2).
a(0) = 0, a(1) = 1, a(2) = 2, a(n) = 2^(A072649(n) - 1) + a(n - A000045(1 + A072649(n))). - Antti Karttunen
It appears that this sequence gives m such that A082759(3*m) is odd; or, probably equivalently, m such that A037011(3*m) = 1. - Benoit Cloitre, Jun 20 2003
If m is in the sequence then so are 2*m and 4*m + 1. - Henry Bottomley, Jan 11 2005
A116361(a(n)) <= 1. - Reinhard Zumkeller, Feb 04 2006
A085357(a(n)) = 1; A179821(a(n)) = a(n). - Reinhard Zumkeller, Jul 31 2010
a(n)/n^k is bounded (but does not tend to a limit), where k = 1.44... = A104287. - Charles R Greathouse IV, Sep 19 2012
a(n) = a(A193564(n+1))*2^(A003849(n) + 1) + A003849(n) for n > 0. - Daniel Starodubtsev, Aug 05 2021
There are Fibonacci(n+1) terms with up to n bits in this sequence. - Charles R Greathouse IV, Oct 22 2021
Sum_{n>=1} 1/a(n) = 3.704711752910469457886531055976801955909489488376627037756627135425780134020... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 12 2022

Edited by Antti Karttunen, Feb 21 2006
Cross reference to A007820 added (into O-Y.C. comment) by Jason Kimberley, Sep 14 2009
Typo corrected by Jeffrey Shallit, Sep 26 2014

A048716 Numbers n such that binary expansion matches ((0)00(1?)1)(0*).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 17, 18, 19, 24, 25, 32, 33, 34, 35, 36, 38, 48, 49, 50, 51, 64, 65, 66, 67, 68, 70, 72, 73, 76, 96, 97, 98, 99, 100, 102, 128, 129, 130, 131, 132, 134, 136, 137, 140, 144, 145, 146, 147, 152, 153, 192, 193, 194, 195, 196, 198, 200, 201

Offset: 1

Antti Karttunen, Mar 30 1999

If bit i is 1, then bits i+-2 must be 0. All terms satisfy A048725(n) = 5*n.
It appears that n is in the sequence if and only if C(5n,n) is odd (cf. A003714). - Benoit Cloitre, Mar 09 2003
Yes, as remarked in A048715, "This is easily proved using the well-known result that the multiplicity with which a prime p divides C(n+m,n) is the number of carries when adding n+m in base p." - Jason Kimberley, Dec 21 2011
A116361(a(n)) <= 2. - Reinhard Zumkeller, Feb 04 2006

Superset of A048715 and A048719. Union of A004742 and A003726.

Cf. A048729, A003714, A115845, A115847, A116360.

Reap[Do[If[OddQ[Binomial[5n, n]], Sow[n]], {n, 0, 400}]][[2, 1]]
(* Second program: *)
filterQ[n_] := With[{bb = IntegerDigits[n, 2]}, MatchQ[bb, {0}|{1}|{1, 1}|{_, 0, , 1, __}|{_ 1, , 0, __}] && !MatchQ[bb, {_, 1, , 1, __}]];
Select[Range[0, 201], filterQ] (* Jean-François Alcover, Dec 31 2020 *)

is(n)=!bitand(n,n>>2) \\ Charles R Greathouse IV, Oct 03 2016

list(lim)=my(v=List(),n,t); while(n<=lim, t=bitand(n,n>>2); if(t, n+=1<Charles R Greathouse IV, Oct 22 2021

A061858 Differences between the ordinary multiplication table A004247 and the carryless multiplication table for GF(2)[X] polynomials A048720, i.e., the effect of the carry bits in binary multiplication.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 8, 0, 0, 0, 0, 0, 0, 12, 0, 8, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 16, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 24, 0, 0, 0

Offset: 0

Antti Karttunen, May 11 2001

			From _Peter Munn_, Jan 28 2021: (Start)
The top left 12 X 12 corner of the table:
      |  0   1   2   3   4   5   6   7   8   9  10  11
------+------------------------------------------------
   0  |  0   0   0   0   0   0   0   0   0   0   0   0
   1  |  0   0   0   0   0   0   0   0   0   0   0   0
   2  |  0   0   0   0   0   0   0   0   0   0   0   0
   3  |  0   0   0   4   0   0   8  12   0   0   0   4
   4  |  0   0   0   0   0   0   0   0   0   0   0   0
   5  |  0   0   0   0   0   8   0   8   0   0  16  16
   6  |  0   0   0   8   0   0  16  24   0   0   0   8
   7  |  0   0   0  12   0   8  24  28   0   0  16  28
   8  |  0   0   0   0   0   0   0   0   0   0   0   0
   9  |  0   0   0   0   0   0   0   0   0  16   0  16
  10  |  0   0   0   0   0  16   0  16   0   0  32  32
  11  |  0   0   0   4   0  16   8  28   0  16  32  52
(End)

"Zoomed in" variant: A061859.
Rows/columns 3, 5 and 7 are given by A048728, A048729, A048730.
Main diagonal divided by 4: A213673.
Numbers that generate no carries when multiplied in binary by 11_2: A003714, by 101_2: A048716, by 1001_2: A115845, by 10001_2: A115847, by 100001_2: A114086.
Other sequences related to the presence/absence of a carry in binary multiplication: A116361, A235034, A235040, A236378, A266195, A289726.

a(n) = A004247(n) - A048720(n).

A115847 Integers i such that 17i = 17 X i, i.e., 16i XOR i = 17*i.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 33, 36, 37, 40, 41, 44, 45, 48, 52, 56, 60, 64, 65, 66, 67, 72, 73, 74, 75, 80, 82, 88, 90, 96, 97, 104, 105, 112, 120, 128, 129, 130, 131, 132, 133, 134, 135, 144, 146, 148, 150

Offset: 0

Antti Karttunen, Feb 01 2006

nonn

Here * stands for ordinary multiplication and X means carryless (GF(2)[X]) multiplication (A048720).

A116361(a(n)) <= 4. - Reinhard Zumkeller, Feb 04 2006

Cf. A115848 shows this sequence in binary. Complement of A115849. Differs from A032966 for the first time at n=25, where A032966(25)=34 while a(25)=33.

Cf. A003714, A048716, A115845, A116360.

is(n)=bitxor(n,16*n)==17*n \\ Charles R Greathouse IV, Dec 08 2013

A115845 Numbers n such that there is no bit position where the binary expansions of n and 8n are both 1.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 17, 20, 21, 24, 28, 32, 33, 34, 35, 40, 42, 48, 49, 56, 64, 65, 66, 67, 68, 69, 70, 71, 80, 81, 84, 85, 96, 97, 98, 99, 112, 113, 128, 129, 130, 131, 132, 133, 134, 135, 136, 138, 140, 142, 160, 161, 162, 163, 168, 170, 192

Offset: 1

Antti Karttunen, Feb 01 2006

nonn

Equivalently, numbers n such that 9*n = 9 X n, i.e., 8*n XOR n = 9*n. Here * stands for ordinary multiplication and X means carryless (GF(2)[X]) multiplication (A048720).
Equivalently, numbers n such that the binomial coefficient C(9n,n) (A169958) is odd. - Zak Seidov, Aug 06 2010
The equivalence of these three definitions follows from Lucas's theorem on binomial coefficients. - N. J. A. Sloane, Sep 01 2010
Clearly all numbers k*2^i for 1 <= k <= 7 have this property. - N. J. A. Sloane, Sep 01 2010
A116361(a(n)) <= 3. - Reinhard Zumkeller, Feb 04 2006

N. J. A. Sloane and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences defined by congruent products between domains N and GF(2)[X]
Index entries for sequences defined by congruent products under XOR

A115846 shows this sequence in binary.
A033052 is a subsequence.
Cf. A003714, A048716, A115847, A116360, A005809, A003714, A048716, A048715.

Reap[Do[If[OddQ[Binomial[9n,n]],Sow[n]],{n,0,400}]][[2,1]] (* Zak Seidov, Aug 06 2010 *)

is(n)=!bitand(n,n<<3) \\ Charles R Greathouse IV, Sep 23 2012

a(n)/n^k is bounded (but does not tend to a limit), where k = 1.44... = A104287. - Charles R Greathouse IV, Sep 23 2012

Edited with a new definition by N. J. A. Sloane, Sep 01 2010, merging this sequence with a sequence submitted by Zak Seidov, Aug 06 2010

A261891 Least k>0 such that n AND (k*n) = 0, where AND stands for the binary AND operator.

Original entry on oeis.org

2, 2, 4, 2, 2, 4, 8, 2, 2, 2, 12, 4, 10, 8, 16, 2, 2, 2, 4, 2, 2, 12, 24, 4, 4, 10, 12, 8, 10, 16, 32, 2, 2, 2, 4, 2, 2, 4, 40, 2, 2, 2, 12, 12, 10, 24, 48, 4, 4, 4, 4, 10, 34, 12, 56, 8, 18, 10, 12, 16, 42, 32, 64, 2, 2, 2, 4, 2, 2, 4, 8, 2, 2, 2, 12, 4, 10

Offset: 1

Paul Tek, Sep 05 2015

All terms are even.
a(A003714(n)) = 2 for any n>0.
a(A004780(n)) > 2 for any n>0.
a(n) <= 2^A116361(n) for any n>0.
a(2n) = a(n) for any n>0.

			For n=7:
+---+-------------+
| k | 7 AND (k*7) |
|   | (in binary) |
+---+-------------+
| 1 |         111 |
| 2 |         110 |
| 3 |         101 |
| 4 |         100 |
| 5 |          11 |
| 6 |          10 |
| 7 |           1 |
| 8 |           0 |
+---+-------------+
Hence, a(7) = 8.

Paul Tek, Table of n, a(n) for n = 1..16384

Cf. A003714, A004780, A116361, A261892.

Table[k = 1; While[BitAnd[k n, n] != 0, k++]; k, {n, 60}] (* Michael De Vlieger, Sep 06 2015 *)

a(n) = {k=1; while (bitand(n, k*n), k++); k;} \\ Michel Marcus, Sep 06 2015

sub a {
   my $n = shift;
   my $k = 1;
   while ($n & ($k*$n)) {
      $k++;
   }
   return $k;
}

from itertools import count
def A261891(n): return next(k for k in count(2) if not n&k*n) # Chai Wah Wu, Jul 19 2024

A114086 Numbers m such that m XOR 32m = 33m.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 65, 68, 69, 72, 73, 76, 77, 80, 81, 84, 85, 88, 89, 92, 93, 96, 100, 104, 108, 112, 116, 120

Offset: 0

Reinhard Zumkeller, Feb 04 2006

nonn

A116361(a(n)) <= 5.

Index entries for sequences defined by congruent products under XOR

Differs from A001477 for the first time at n=33 (33, 35, 37, 39, etc. are not present in this sequence). Cf. A003714, A048716, A115845, A115847.

is(n)=bitxor(n, 32*n)==33*n \\ Charles R Greathouse IV, Sep 25 2024

cp's OEIS Frontend

A116362 Smallest m such that A116361(m) = n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A003714 Fibbinary numbers: if n = F(i1) + F(i2) + ... + F(ik) is the Zeckendorf representation of n (i.e., write n in Fibonacci number system) then a(n) = 2^(i1 - 2) + 2^(i2 - 2) + ... + 2^(ik - 2). Also numbers whose binary representation contains no two adjacent 1's.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Python

Python

Python

Scala

Formula

Extensions

A048716 Numbers n such that binary expansion matches ((0)*00(1?)1)*(0*).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A061858 Differences between the ordinary multiplication table A004247 and the carryless multiplication table for GF(2)[X] polynomials A048720, i.e., the effect of the carry bits in binary multiplication.

Views

Author

Keywords

Examples

Crossrefs

Formula

A115847 Integers i such that 17*i = 17 X i, i.e., 16*i XOR i = 17*i.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A115845 Numbers n such that there is no bit position where the binary expansions of n and 8n are both 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A261891 Least k>0 such that n AND (k*n) = 0, where AND stands for the binary AND operator.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

A048716 Numbers n such that binary expansion matches ((0)00(1?)1)(0*).

A115847 Integers i such that 17i = 17 X i, i.e., 16i XOR i = 17*i.

A114086 Numbers m such that m XOR 32m = 33m.