A268717 -id:A268717 - cp's OEIS frontend

A092246 Odd "odious" numbers (A000069).

1, 7, 11, 13, 19, 21, 25, 31, 35, 37, 41, 47, 49, 55, 59, 61, 67, 69, 73, 79, 81, 87, 91, 93, 97, 103, 107, 109, 115, 117, 121, 127, 131, 133, 137, 143, 145, 151, 155, 157, 161, 167, 171, 173, 179, 181, 185, 191, 193, 199, 203, 205, 211, 213, 217, 223, 227, 229, 233

Offset: 1

Benoit Cloitre, Feb 23 2004

In other words, numbers having a binary representation ending in 1, and an odd number of 1's overall. It follows that by decrementing an odd odious number, one gets an even evil number (A125592). - Ralf Stephan, Aug 27 2013
The members of the sequence may be called primitive odious numbers because every odious number is a power of 2 times one of these numbers. Note that the difference between consecutive terms is either 2, 4, or 6. - T. D. Noe, Jun 06 2007
From Gary W. Adamson, Apr 06 2010: (Start)
a(n) = A026147(n)-th odd number, where A026147 = (1, 4, 6, 7, 10, 11, ...); e.g.,
         n: 1   2   3   4   5   6   7   8   9  10  11
  n-th odd: 1   3   5   7   9  11  13  15  17  19  21
      a(n): 1           7      11  13          19  21
etc. (End)
Numbers m, such that when merge-sorting lists of length m, the maximal number of comparisons is even: A003071(a(n)) = A230720(n). - Reinhard Zumkeller, Oct 28 2013
Fixed points of permutation pair A268717/A268718. - Antti Karttunen, Feb 29 2016

Cf. A129771, A026147.
Cf. A230709 (complement).
Cf. A268717, A268718, A268673.

a092246 n = a092246_list !! (n - 1)
a092246_list = filter odd a000069_list
-- Reinhard Zumkeller, Oct 28 2013

Table[If[n < 1, 0, 2 n - 1 - Mod[First@ DigitCount[n - 1, 2], 2]], {n, 120}] /. n_ /; EvenQ@ n -> Nothing (* Michael De Vlieger, Feb 29 2016 *)
Select[Range[1, 1001, 2], OddQ[Total[IntegerDigits[#, 2]]]&] (* Jean-François Alcover, Mar 15 2016 *)

is(n)=n%2&&hammingweight(n)%2 \\ Charles R Greathouse IV, Mar 21 2013

a(n)=4*n-if(hammingweight(n-1)%2,1,3) \\ Charles R Greathouse IV, Mar 22 2013

def A092246(n): return (n<<2)-(1 if (n-1).bit_count()&1 else 3) # Chai Wah Wu, Mar 03 2023

a(n) = 4*n + 2*A010060(n-1) - 3;

a(n) = 2*A001969(n-1) + 1.

A268718 Permutation of natural numbers: a(0) = 0, a(n) = 1 + A003188(A006068(n)-1), where A003188 is binary Gray code and A006068 is its inverse.

Original entry on oeis.org

0, 1, 4, 2, 6, 8, 3, 7, 10, 12, 15, 11, 5, 13, 16, 14, 18, 20, 23, 19, 29, 21, 24, 22, 9, 25, 28, 26, 30, 32, 27, 31, 34, 36, 39, 35, 45, 37, 40, 38, 57, 41, 44, 42, 46, 48, 43, 47, 17, 49, 52, 50, 54, 56, 51, 55, 58, 60, 63, 59, 53, 61, 64, 62, 66, 68, 71, 67, 77, 69, 72, 70, 89, 73, 76, 74, 78, 80, 75, 79, 113, 81

Offset: 0

Antti Karttunen, Feb 12 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Inverse: A268717.
Cf. A003188, A006068, A010060, A105081, A128309, A166519, A171977, A277822.
Row 1 of array A268830.
Cf. A092246 (fixed points).
Cf. A268818 ("square" of this permutation).
Cf. A268822 ("shifted square"), A268824 ("shifted cube") and also A268826, A268828 and A268832 (higher "shifted powers").

{0}~Join~Table[1 + BitXor[#, Floor[#/2]] &[BitXor @@ Table[Floor[n/2^m], {m, 0, Floor[Log[2, n]]}] - 1], {n, 81}] (* Michael De Vlieger, Feb 29 2016, after Jean-François Alcover at A006068 and Robert G. Wilson v at A003188 *)

a003188(n)=bitxor(n, n>>1);
a006068(n)= {
    my( s=1, ns );
    while ( 1,
        ns = n >> s;
        if ( 0==ns, break() );
        n = bitxor(n, ns);
        s <<= 1;
    );
    return (n);
} \\ by Joerg Arndt
a(n)=if(n==0, 0, 1 + a003188(a006068(n) - 1)); \\ Indranil Ghosh, Jun 07 2017

def a003188(n): return n^(n>>1)
def a006068(n):
    s=1
    while True:
        ns=n>>s
        if ns==0: break
        n=n^ns
        s<<=1
    return n
def a(n): return 0 if n==0 else 1 + a003188(a006068(n) - 1) # Indranil Ghosh, Jun 07 2017

(define (A268718 n) (if (zero? n) n (A105081 (A006068 n))))

a(0) = 0, and for n >= 1, a(n) = A105081(A006068(n)) = 1 + A003188(A006068(n)-1).
Other identities. For all n >= 1:
a(A128309(n)) = A128309(n)+2. [Maps any even odious number to that number + 2.]
From Alan Michael Gómez Calderón, May 29 2025: (Start)
a(n) - 1 = A268717(n+1) XOR (A171977(n)+1) for n >= 1;
a(2*n-1) - 1 = (2-A010060(n-1)) XOR (A166519(n-1)-1) for n >= 1;
a(2*n) - 1 = (a(2*(n+1)-1)-1) XOR 2^A277822(n) for n >= 1. (End)

A268715 Square array A(i,j) = A003188(A006068(i) + A006068(j)), read by antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 6, 6, 3, 4, 2, 5, 2, 4, 5, 12, 7, 7, 12, 5, 6, 4, 15, 6, 15, 4, 6, 7, 7, 13, 13, 13, 13, 7, 7, 8, 5, 4, 12, 9, 12, 4, 5, 8, 9, 24, 12, 5, 11, 11, 5, 12, 24, 9, 10, 8, 27, 4, 14, 10, 14, 4, 27, 8, 10, 11, 11, 25, 25, 10, 15, 15, 10, 25, 25, 11, 11, 12, 9, 8, 24, 29, 14, 12, 14, 29, 24, 8, 9, 12, 13, 13, 24, 9, 31, 31, 13, 13, 31, 31, 9, 24, 13, 13

Offset: 0

Antti Karttunen, Feb 12 2016

Each row n is row A006068(n) of array A268820 without its A006068(n) initial terms.

			The top left [0 .. 15] x [0 .. 15] section of the array:
   0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15
   1,  3,  6,  2, 12,  4,  7,  5, 24,  8, 11,  9, 13, 15, 10, 14
   2,  6,  5,  7, 15, 13,  4, 12, 27, 25,  8, 24, 14, 10,  9, 11
   3,  2,  7,  6, 13, 12,  5,  4, 25, 24,  9,  8, 15, 14, 11, 10
   4, 12, 15, 13,  9, 11, 14, 10, 29, 31, 26, 30,  8, 24, 27, 25
   5,  4, 13, 12, 11, 10, 15, 14, 31, 30, 27, 26,  9,  8, 25, 24
   6,  7,  4,  5, 14, 15, 12, 13, 26, 27, 24, 25, 10, 11,  8,  9
   7,  5, 12,  4, 10, 14, 13, 15, 30, 26, 25, 27, 11,  9, 24,  8
   8, 24, 27, 25, 29, 31, 26, 30, 17, 19, 22, 18, 28, 20, 23, 21
   9,  8, 25, 24, 31, 30, 27, 26, 19, 18, 23, 22, 29, 28, 21, 20
  10, 11,  8,  9, 26, 27, 24, 25, 22, 23, 20, 21, 30, 31, 28, 29
  11,  9, 24,  8, 30, 26, 25, 27, 18, 22, 21, 23, 31, 29, 20, 28
  12, 13, 14, 15,  8,  9, 10, 11, 28, 29, 30, 31, 24, 25, 26, 27
  13, 15, 10, 14, 24,  8, 11,  9, 20, 28, 31, 29, 25, 27, 30, 26
  14, 10,  9, 11, 27, 25,  8, 24, 23, 21, 28, 20, 26, 30, 29, 31
  15, 14, 11, 10, 25, 24,  9,  8, 21, 20, 29, 28, 27, 26, 31, 30

Antti Karttunen, Table of n, a(n) for n = 0..15050; the first 173 antidiagonals of the array

Cf. A003188, A006068, A268714, A268820.
Main diagonal: A001969.
Row 0, column 0: A001477.
Row 1, column 1: A268717.
Antidiagonal sums: A268837.
Cf. A268719 (the lower triangular section).
Cf. also A268725.

A003188[n_] := BitXor[n, Floor[n/2]]; A006068[n_] := BitXor @@ Table[Floor[ n/2^m], {m, 0, Log[2, n]}]; A006068[0]=0; A[i_, j_] := A003188[A006068[i] + A006068[j]]; Table[A[i-j, j], {i, 0, 13}, {j, 0, i}] // Flatten (* Jean-François Alcover, Feb 17 2016 *)

def a003188(n): return n^(n>>1)
def a006068(n):
    s=1
    while True:
        ns=n>>s
        if ns==0: break
        n=n^ns
        s<<=1
    return n
def T(n, k): return a003188(a006068(n) + a006068(k))
for n in range(21): print([T(n - k, k) for k in range(n + 1)]) # Indranil Ghosh, Jun 07 2017

(define (A268715 n) (A268715bi (A002262 n) (A025581 n)))
(define (A268715bi row col) (A003188 (+ (A006068 row) (A006068 col))))
;; Alternatively, extracting data from array A268820:
(define (A268715bi row col) (A268820bi (A006068 row) (+ (A006068 row) col)))

A(i,j) = A003188(A006068(i) + A006068(j)) = A003188(A268714(i,j)).

A(row,col) = A268820(A006068(row), (A006068(row)+col)).

A066194 A permutation of the integers (a fractal sequence): a(n) = A006068(n-1) + 1.

Original entry on oeis.org

1, 2, 4, 3, 8, 7, 5, 6, 16, 15, 13, 14, 9, 10, 12, 11, 32, 31, 29, 30, 25, 26, 28, 27, 17, 18, 20, 19, 24, 23, 21, 22, 64, 63, 61, 62, 57, 58, 60, 59, 49, 50, 52, 51, 56, 55, 53, 54, 33, 34, 36, 35, 40, 39, 37, 38, 48, 47, 45, 46, 41, 42, 44, 43, 128, 127, 125, 126, 121, 122

Offset: 1

Wouter Meeussen, Dec 15 2001

With an initial zero, inverse of the Gray Code (A003188). See also A006068. - Robert G. Wilson v, Jun 22 2014

I suspect the above comment refers to function A105081(n) = 1 + A003188(n - 1), n >= 1. - Antti Karttunen, Feb 15 2016

			Third nesting gives {1,2,4,3, 8,7,5,6} by means of joining the lists {1,2,4,3} = second nesting and {8,7,6,5} permuted by {1,2,4,3} giving {8,7,5,6}.

Antti Karttunen, Table of n, a(n) for n = 1..8192 (first 1024 terms from Robert G. Wilson v)
Index entries for sequences that are permutations of the natural numbers

Inverse: A105081.

Cf. A003188, A006068, A268717.

Nest[ Join[ #, (Length[ #] + Range[ Length[ #], 1, -1 ])[[ # ]]] &, {1}, 7 ]
GrayCode[n_] := BitXor[n, Floor[n/2]]; t = Array[ GrayCode, 1000, 0]; Table[ Position[ t, n], {n, 0, 100}] // Flatten (* Robert G. Wilson v, Jun 22 2014 *)

def A066194(n):
    k, m = n-1, n-1>>1
    while m > 0:
        k ^= m
        m >>= 1
    return k+1 # Chai Wah Wu, Jul 01 2022

(define (A066194 n) (+ 1 (A006068 (- n 1)))) ;; Antti Karttunen, Feb 14 2016

a(n) = A006068(n-1) + 1, n >= 1. - Philippe Deléham, Apr 29 2005
a(n) = A006068(A268717(n)), composition of related permutations. - Antti Karttunen, Feb 14 2016
a(n) = 1 + Sum_{j=1..n-1} (1/6)*(-3 + (-1)^A007814(j) + 2^(A007814(j) + 3))*(-1)^(A000120(j) + 1). - John Erickson, Oct 18 2018

Deléham's formula added to the name by Antti Karttunen, Feb 14 2016

A302027 Permutation of nonnegative integers: a(0) = 0; for n >= 1, a(n) = A057889(1+A057889(n-1)), where A057889 is a bijective bit-reverse.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 11, 12, 15, 16, 17, 18, 25, 22, 21, 26, 27, 30, 19, 20, 29, 28, 23, 24, 31, 32, 33, 34, 49, 38, 41, 42, 51, 46, 37, 50, 53, 54, 43, 58, 55, 62, 35, 36, 57, 44, 45, 52, 59, 60, 39, 40, 61, 56, 47, 48, 63, 64, 65, 66, 97, 70, 81, 74, 99, 78, 73, 82, 101, 86, 83, 90, 103, 94, 69, 98, 105, 102, 85

Offset: 0

Antti Karttunen, Apr 26 2018

Antti Karttunen, Table of n, a(n) for n = 0..16383
David Newman, et al., New sequences from old, Discussion on SeqFan-mailing list, January 2018.
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A302028 (inverse).
Cf. A057889.
Cf. also A268717, A302793.

f[n_] := FromDigits[Reverse[IntegerDigits[n, 2]], 2]*2^IntegerExponent[n, 2]; Fold[Append[#1, f[1 + f[#2 - 1]]] &, {0, 1}, Range[2, 85]] (* Michael De Vlieger, Apr 27 2018, after Ivan Neretin at A057889 *)

A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A057889(n) = if(!n,n,A030101(n/(2^valuation(n,2))) * (2^valuation(n, 2)));
A302027(n) = if(!n,n,A057889(1+A057889(n-1)));

a(0) = 0; for n >= 1, a(n) = A057889(1+A057889(n-1)).

A302793 Permutation of nonnegative integers: a(0) = 0; for n >= 1, a(n) = A193231(1+A193231(n-1)), where A193231(n) is blue code of n.

Original entry on oeis.org

0, 1, 3, 5, 2, 6, 4, 7, 15, 17, 8, 11, 9, 13, 10, 12, 14, 18, 16, 19, 20, 21, 23, 30, 22, 25, 51, 24, 26, 27, 28, 31, 29, 39, 32, 35, 33, 37, 45, 36, 38, 41, 43, 85, 42, 46, 44, 47, 40, 49, 54, 48, 50, 60, 52, 55, 53, 58, 56, 59, 34, 61, 63, 57, 62, 66, 64, 67, 75, 69, 71, 65, 70, 73, 78, 72, 74, 102, 76, 79, 77, 81

Offset: 0

Antti Karttunen, Apr 26 2018

Antti Karttunen, Table of n, a(n) for n = 0..16383
David Newman, et al., New sequences from old, Discussion on SeqFan-mailing list, January 2018.
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A302794 (inverse).
Cf. A193231, A302795.
Cf. also A268717, A302027.

A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ From A193231
A302793(n) = if(!n,n,A193231(1+A193231(n-1)));

a(0) = 0; for n >= 1, a(n) = A193231(1+A193231(n-1)).

A268673 a(0) = 0; a(1) = 1; for n > 1, a(n) = 1 + 4*A092246(n-1).

Original entry on oeis.org

0, 1, 5, 29, 45, 53, 77, 85, 101, 125, 141, 149, 165, 189, 197, 221, 237, 245, 269, 277, 293, 317, 325, 349, 365, 373, 389, 413, 429, 437, 461, 469, 485, 509, 525, 533, 549, 573, 581, 605, 621, 629, 645, 669, 685, 693, 717, 725, 741, 765, 773, 797, 813, 821, 845, 853, 869, 893, 909, 917, 933, 957, 965, 989, 1005

Offset: 0

Antti Karttunen, Feb 19 2016

nonn

Seems to be also the fixed points of permutations A268823 and A268824.

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A268823, A268824.
Cf. also A092246, A268717.
Many (but not all) terms of A013710 seem to be included.

Join[{0, 1}, 1 + 4 Select[Range[1, 251, 2], OddQ[Total[IntegerDigits[#, 2]]]&]] (* Jean-François Alcover, Mar 15 2016 *)

def A268673(n): return (((m:=n-2)<<4)+(13 if m.bit_count()&1 else 5)) if n>1 else n # Chai Wah Wu, Mar 03 2023

(define (A268673 n) (if (<= n 1) n (+ 1 (* 4 (A092246 (- n 1))))))

a(0) = 0; a(1) = 1; for n > 1, a(n) = 1 + 4*A092246(n-1).

A268835 Main diagonal of arrays A268833 & A268834: a(n) = A101080(n, A268820(n, 2*n)).

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 2, 3, 2, 3, 4, 5, 2, 1, 4, 3, 2, 3, 4, 5, 4, 3, 6, 5, 2, 3, 2, 1, 4, 5, 4, 3, 2, 3, 4, 5, 4, 3, 6, 5, 4, 5, 4, 3, 6, 7, 6, 5, 2, 3, 4, 3, 2, 3, 2, 3, 4, 5, 6, 5, 4, 3, 4, 3, 2, 3, 4, 5, 4, 3, 6, 5, 4, 5, 4, 3, 6, 7, 6, 5, 4, 5, 6, 5, 4, 5, 4, 5, 6, 7, 8, 7, 6, 5, 6, 5, 2, 3, 4, 3, 4, 5, 4, 5, 2, 3, 4, 5, 2, 1, 4, 3, 4, 5, 6, 5, 6, 5, 6, 5, 4

Offset: 0

Antti Karttunen, Feb 15 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1024
Indranil Ghosh, C program to generate the sequence

Cf. A101080, A268820, A268833, A268834.

A101080[n_, k_]:= DigitCount[BitXor[n, k], 2, 1];A003188[n_]:=BitXor[n, Floor[n/2]]; A006068[n_]:=If[n<2, n, Block[{m=A006068[Floor[n/2]]}, 2m + Mod[Mod[n,2] + Mod[m, 2], 2]]]; a[r_, 0]:= 0; a[0, c_]:=c; a[r_, c_]:= A003188[1 + A006068[a[r - 1, c - 1]]]; Flatten@ Table[A101080[n, a[n, 2n]], {n, 0, 300}] (* Indranil Ghosh, Apr 02 2017 *)

b(n) = if(n<1, 0, b(n\2) + n%2);
A101080(n, k) = b(bitxor(n, k));
A003188(n) = bitxor(n, n\2);
A006068(n) = if(n<2, n, {my(m = A006068(n\2)); 2*m + (n%2 + m%2)%2});
A268820(r, c) = if(r==0, c, if(c==0, 0, A003188(1 + A006068(A268820(r - 1, c - 1)))));
for(n=0, 300, print1(A101080(n, A268820(n, 2*n)),", ")) \\ Indranil Ghosh, Apr 02 2017

def A101080(n, k): return bin(n^k)[2:].count("1")
def A003188(n): return n^(n//2)
def A006068(n):
    if n<2: return n
    else:
        m=A006068(n//2)
        return 2*m + (n%2 + m%2)%2
def A268717(n): return 0 if n<1 else A003188(1 + A006068(n - 1))
def A268820(r, c): return c if r<1 else 0 if c<1 else A003188(1 + A006068(A268820(r - 1, c - 1)))
print([A101080(n, A268820(n, 2*n)) for n in range(301)]) # Indranil Ghosh, Apr 02 2017

(define (A268835 n) (A101080bi n (A268820bi n (* 2 n))))
(define (A268835 n) (A268833bi n n)) ;; Code for A268833bi given in A268833.

a(n) = A101080(n, A268820(n, 2*n)).

cp's OEIS Frontend

A092246 Odd "odious" numbers (A000069).

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A268718 Permutation of natural numbers: a(0) = 0, a(n) = 1 + A003188(A006068(n)-1), where A003188 is binary Gray code and A006068 is its inverse.

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A268715 Square array A(i,j) = A003188(A006068(i) + A006068(j)), read by antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

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A066194 A permutation of the integers (a fractal sequence): a(n) = A006068(n-1) + 1.

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A302027 Permutation of nonnegative integers: a(0) = 0; for n >= 1, a(n) = A057889(1+A057889(n-1)), where A057889 is a bijective bit-reverse.

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A302793 Permutation of nonnegative integers: a(0) = 0; for n >= 1, a(n) = A193231(1+A193231(n-1)), where A193231(n) is blue code of n.

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A268673 a(0) = 0; a(1) = 1; for n > 1, a(n) = 1 + 4*A092246(n-1).

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