A277823 -id:A277823 - cp's OEIS frontend

A129771 Evil odd numbers.

3, 5, 9, 15, 17, 23, 27, 29, 33, 39, 43, 45, 51, 53, 57, 63, 65, 71, 75, 77, 83, 85, 89, 95, 99, 101, 105, 111, 113, 119, 123, 125, 129, 135, 139, 141, 147, 149, 153, 159, 163, 165, 169, 175, 177, 183, 187, 189, 195, 197, 201, 207, 209, 215, 219, 221, 225, 231, 235

Offset: 1

Tanya Khovanova, May 16 2007

A heuristic argument suggests that, as n tends to infinity, a(n)/n converges to 4. - Stefan Steinerberger, May 17 2007
These numbers may be called primitive evil numbers because every evil number is a power of 2 multiplied by one of these numbers. Note that the difference between consecutive terms is either 2, 4, or 6. - T. D. Noe, Jun 06 2007
If m is in the sequence, then so is 2m-1 because in binary, m is x1 and 2m-1 is x01. Presumably the numbers that generate the whole sequence by application of n -> 2n-1 are the evil numbers times 4 plus 3. - Ralf Stephan, May 25 2013

Francisco J. Muñoz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Francisco J. Muñoz and Juan Carlos Nuño, Rule-based Generation of de Bruijn Sequences: Memory and Learning, arXiv:2507.09764 [cs.FL], 2025. See p. 9.

Intersection of A001969 and A005408.
Supersequence of A093688.
Cf. A092246 (odd odious numbers).
Column 2 of A277880, positions of 1's in A277808 (2's in A277822).
Cf. A000069, A128309, A277823, A277902.

Select[Range[300], OddQ[ # ] && EvenQ[DigitCount[ #, 2, 1]] &] (* Stefan Steinerberger, May 17 2007 *)
Select[Range[300], EvenQ[Plus @@ IntegerDigits[ #, 2]] && OddQ[ # ] &]

is(n)=n%2 && hammingweight(n)%2==0 \\ Charles R Greathouse IV, Mar 21 2013

a(n)=4*n-if(hammingweight(n-1)%2,3,1) \\ Charles R Greathouse IV, Mar 21 2013

def A129771(n): return (((m:=n-1)<<1)+(m.bit_count()&1^1)<<1)+1 # Chai Wah Wu, Mar 09 2023

a(n) = 2*A000069(n) + 1. a(n) is 1 plus twice odious numbers.
a(n) = A128309(n) + 1. a(n) is 1 plus odious even numbers.
A132680(a(n)) = A132680((a(n)-1)/2) + 2. - Reinhard Zumkeller, Aug 26 2007
a(n) = 4n + O(1). - Charles R Greathouse IV, Mar 21 2013
a(n) = A001969(1+A000069(n)) = A277902(A277823(n)). - Antti Karttunen, Nov 05 2016

More terms from Stefan Steinerberger, May 17 2007

A277320 Square array A(r,c) = A048720(A065621(r), c), read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

1, 2, 2, 3, 4, 7, 4, 6, 14, 4, 5, 8, 9, 8, 13, 6, 10, 28, 12, 26, 14, 7, 12, 27, 16, 23, 28, 11, 8, 14, 18, 20, 52, 18, 22, 8, 9, 16, 21, 24, 57, 56, 29, 16, 25, 10, 18, 56, 28, 46, 54, 44, 24, 50, 26, 11, 20, 63, 32, 35, 36, 39, 32, 43, 52, 31, 12, 22, 54, 36, 104, 42, 58, 40, 100, 46, 62, 28, 13, 24, 49, 40, 101, 112, 49, 48, 125, 104, 33, 56, 21

Offset: 1

Antti Karttunen, Nov 01 2016

			The top left corner of the array:
   1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12
   2,   4,   6,   8,  10,  12,  14,  16,  18,  20,  22,  24
   7,  14,   9,  28,  27,  18,  21,  56,  63,  54,  49,  36
   4,   8,  12,  16,  20,  24,  28,  32,  36,  40,  44,  48
  13,  26,  23,  52,  57,  46,  35, 104, 101, 114, 127,  92
  14,  28,  18,  56,  54,  36,  42, 112, 126, 108,  98,  72
  11,  22,  29,  44,  39,  58,  49,  88,  83,  78,  69, 116
   8,  16,  24,  32,  40,  48,  56,  64,  72,  80,  88,  96
  25,  50,  43, 100, 125,  86,  79, 200, 209, 250, 227, 172
  26,  52,  46, 104, 114,  92,  70, 208, 202, 228, 254, 184
  31,  62,  33, 124,  99,  66,  93, 248, 231, 198, 217, 132
  28,  56,  36, 112, 108,  72,  84, 224, 252, 216, 196, 144
  21,  42,  63,  84,  65, 126, 107, 168, 189, 130, 151, 252
  22,  44,  58,  88,  78, 116,  98, 176, 166, 156, 138, 232
  19,  38,  53,  76,  95, 106, 121, 152, 139, 190, 173, 212
  16,  32,  48,  64,  80,  96, 112, 128, 144, 160, 176, 192
  49,  98,  83, 196, 245, 166, 151, 392, 441, 490, 475, 332
  50, 100,  86, 200, 250, 172, 158, 400, 418, 500, 454, 344
  55, 110,  89, 220, 235, 178, 133, 440, 399, 470, 481, 356

Transpose: A277199.
Main diagonal: A277699.
Row 1: A000027, Row 3: A048727.
Column 1: A065621, Column 3: A277823, Column 5: A277825.
Cf. A277820 (array obtained by selecting only the columns with an index A001317(k), k=0..).
Cf. A048720, A115872.

(define (A277320 n) (A277320bi (A002260 n) (A004736 n)))
(define (A277320bi row col) (A048720bi (A065621 row) col))

A(r,c) = A048720(A065621(r), c).

A277820 Square array: A(r,1) = A065621(r); for c > 1, A(r,c) = A048724(A(r,c-1)), read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

1, 3, 2, 5, 6, 7, 15, 10, 9, 4, 17, 30, 27, 12, 13, 51, 34, 45, 20, 23, 14, 85, 102, 119, 60, 57, 18, 11, 255, 170, 153, 68, 75, 54, 29, 8, 257, 510, 427, 204, 221, 90, 39, 24, 25, 771, 514, 765, 340, 359, 238, 105, 40, 43, 26, 1285, 1542, 1799, 1020, 937, 306, 187, 120, 125, 46, 31, 3855, 2570, 2313, 1028, 1275, 854, 461, 136, 135, 114, 33, 28

Offset: 1

Antti Karttunen, Nov 01 2016

For all n >= 1, A277818 (= A268389(n)+1) gives the (one-based) index of the column where n is located in this array, while A268671(n) gives the (one-based) index of the row where it is on.

This array is obtained when one selects from A277320 the columns 1, 3, 5, 15, 17, 51, ..., i.e., those with an index A001317(k).

			The top left corner of the array:
   1,  3,   5,  15,  17,   51,   85,  255,   257,   771,  1285,  3855
   2,  6,  10,  30,  34,  102,  170,  510,   514,  1542,  2570,  7710
   7,  9,  27,  45, 119,  153,  427,  765,  1799,  2313,  6939, 11565
   4, 12,  20,  60,  68,  204,  340, 1020,  1028,  3084,  5140, 15420
  13, 23,  57,  75, 221,  359,  937, 1275,  3341,  5911, 14649, 19275
  14, 18,  54,  90, 238,  306,  854, 1530,  3598,  4626, 13878, 23130
  11, 29,  39, 105, 187,  461,  599, 1785,  2827,  7453, 10023, 26985
   8, 24,  40, 120, 136,  408,  680, 2040,  2056,  6168, 10280, 30840
  25, 43, 125, 135, 393,  667, 1965, 2295,  6425, 11051, 32125, 34695
  26, 46, 114, 150, 442,  718, 1874, 2550,  6682, 11822, 29298, 38550
  31, 33,  99, 165, 495,  561, 1619, 2805,  7967,  8481, 25443, 42405
  28, 36, 108, 180, 476,  612, 1708, 3060,  7196,  9252, 27756, 46260
  21, 63,  65, 195, 325,  975, 1105, 3315,  5397, 16191, 16705, 50115
  22, 58,  78, 210, 374,  922, 1198, 3570,  5654, 14906, 20046, 53970
  19, 53,  95, 225, 291,  869, 1455, 3825,  4883, 13621, 24415, 57825
  16, 48,  80, 240, 272,  816, 1360, 4080,  4112, 12336, 20560, 61680
  49, 83, 245, 287, 801, 1379, 4005, 4335, 12593, 21331, 62965, 73247
  50, 86, 250, 270, 786, 1334, 3930, 4590, 12850, 22102, 64250, 69390
  55, 89, 235, 317, 839, 1481, 3675, 4845, 14135, 22873, 60395, 80957

Inverse permutation: A277821.
Transpose: A277819.
Cf. A048724, A065621.
Row 1: A001317.
Column 1: A065621, column 2: A277823, column 3: A277825.
Cf. A268389, A277818, A268671.
Cf. also A193231, A277320, A277810, A276618 (A099884), A248513.
Other related tables or permutations: A277880, A277901.

(define (A277820 n) (A277820bi (A002260 n) (A004736 n)))
(define (A277820bi row col) (if (= 1 col) (A065621 row) (A048724 (A277820bi row (- col 1)))))

A(r,1) = A065621(r); for c > 1, A(r,c) = A048724(A(r,c-1)).
A(r,c) = A048675(A277810(r,c)).
As a composition of other permutations:
a(n) = A277901(A277880(n)).

A277825 a(n) = A048725(A065621(n)) = A048720(A065621(n),5).

Original entry on oeis.org

5, 10, 27, 20, 57, 54, 39, 40, 125, 114, 99, 108, 65, 78, 95, 80, 245, 250, 235, 228, 201, 198, 215, 216, 141, 130, 147, 156, 177, 190, 175, 160, 485, 490, 507, 500, 473, 470, 455, 456, 413, 402, 387, 396, 417, 430, 447, 432, 277, 282, 267, 260, 297, 294, 311, 312, 365, 354, 371, 380, 337, 350, 335, 320, 965, 970, 987, 980, 1017, 1014, 999, 1000

Offset: 1

Antti Karttunen, Nov 02 2016

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences related to binary expansion of n

Column 3 of A277820, Column 5 of A277320.

Cf. A048720, A048724, A048725, A065621, A277823.

def A277825(n): return (m:=n^ (n&~-n)<<1)^m<<2 # Chai Wah Wu, Jun 29 2022

(define (A277825 n) (A048724 (A277823 n)))
(define (A277825 n) (A048720bi (A065621 n) 5)) ;; For A048720bi see the Scheme-program attachment in A048720.

a(n) = A048724(A277823(n)) = A048725(A065621(n)).

a(n) = A048720(A065621(n),5).

A277902 If A010060(n) = 1, a(n) = A000069(A268671(n)), otherwise a(n) = A001969(1+a(A006068(n)/2)).

Original entry on oeis.org

1, 2, 3, 7, 6, 5, 4, 14, 9, 10, 13, 15, 8, 11, 12, 31, 24, 23, 28, 30, 25, 26, 17, 29, 16, 19, 18, 22, 27, 20, 21, 62, 43, 40, 61, 45, 56, 59, 54, 58, 49, 50, 33, 55, 36, 39, 52, 63, 32, 35, 48, 38, 57, 46, 37, 47, 34, 53, 44, 60, 41, 42, 51, 127, 102, 85, 124, 120, 121, 122, 83, 95, 112, 115, 68, 118, 89, 106

Offset: 1

Antti Karttunen, Nov 03 2016

a(n) gives the number that is in the same position in array A277880 as where n is located in array A277820.

			The top left corner of array A277820 is:
   1,  3,  5, 15
   2,  6, 10, 30
   7,  9, 27, 45
   4, 12, 20, 60
  13, 23, 57, 75
while the top left corner of A277880 is:
   1,  3,  6, 12
   2,  5, 10, 20
   4,  9, 18, 36
   7, 15, 30, 60
   8, 17, 34, 68
thus for example, a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 7, a(5) = 6, a(6) = 5, a(7) = 4, a(9) = 9, a(12) = 15, a(13) = 8 and a(27) = 18.

Inverse: A277901.
Cf. A000069, A001969, A001317, A003945, A006068, A010060, A065621, A129771, A268671, A277823, A277825.
Related permutations and arrays: A277820, A277821, A277880.

If A010060(n) = 1 [when n is one of the odious numbers, A000069], then a(n) = A000069(A268671(n)), otherwise a(n) = A001969(1+a(A006068(n)/2)).
As a composition of other permutations:
a(n) = A277880(A277821(n)).
Other identities. For all n >= 1:
A010060(a(n)) = A010060(n). [Preserves the parity of binary weight.]
a(A001317(n)) = A003945(n).
a(A065621(n)) = A000069(n).
a(A277823(n)) = A129771(n).
a(A277825(n)) = 2*A129771(n).

A277819 Transpose of square array A277820.

Original entry on oeis.org

1, 2, 3, 7, 6, 5, 4, 9, 10, 15, 13, 12, 27, 30, 17, 14, 23, 20, 45, 34, 51, 11, 18, 57, 60, 119, 102, 85, 8, 29, 54, 75, 68, 153, 170, 255, 25, 24, 39, 90, 221, 204, 427, 510, 257, 26, 43, 40, 105, 238, 359, 340, 765, 514, 771, 31, 46, 125, 120, 187, 306, 937, 1020, 1799, 1542, 1285, 28, 33, 114, 135, 136, 461, 854, 1275, 1028, 2313, 2570, 3855

Offset: 1

Antti Karttunen, Nov 01 2016

See A277820.

			The top left 10 x 10 corner of the array:
    1,    2,    7,    4,   13,   14,   11,    8,    25,    26
    3,    6,    9,   12,   23,   18,   29,   24,    43,    46
    5,   10,   27,   20,   57,   54,   39,   40,   125,   114
   15,   30,   45,   60,   75,   90,  105,  120,   135,   150
   17,   34,  119,   68,  221,  238,  187,  136,   393,   442
   51,  102,  153,  204,  359,  306,  461,  408,   667,   718
   85,  170,  427,  340,  937,  854,  599,  680,  1965,  1874
  255,  510,  765, 1020, 1275, 1530, 1785, 2040,  2295,  2550
  257,  514, 1799, 1028, 3341, 3598, 2827, 2056,  6425,  6682
  771, 1542, 2313, 3084, 5911, 4626, 7453, 6168, 11051, 11822

Transpose: A277820.
Row 1: A065621, row 2: A277823.
Column 1: A001317.
Cf. A048724, A099884.

(define (A277819 n) (A277820bi (A004736 n) (A002260 n))) ;; Code for A277820bi given in A277820.

A277901 If A010060(n) = 1, a(n) = A065621(A115384(n)), otherwise a(n) = A048724(a(floor(n/2))).

Original entry on oeis.org

1, 2, 3, 7, 6, 5, 4, 13, 9, 10, 14, 15, 11, 8, 12, 25, 23, 27, 26, 30, 31, 28, 18, 17, 21, 22, 29, 19, 24, 20, 16, 49, 43, 57, 50, 45, 55, 52, 46, 34, 61, 62, 33, 59, 36, 54, 56, 51, 41, 42, 63, 47, 58, 39, 44, 37, 53, 40, 38, 60, 35, 32, 48, 97, 83, 125, 98, 75, 103, 100, 86, 119, 109, 110, 89, 107, 92, 114

Offset: 1

Antti Karttunen, Nov 03 2016

a(n) gives the number that is in the same position in array A277820 as where n is located in array A277880.

			The top left corner of array A277880 is:
   1,  3,  6, 12
   2,  5, 10, 20
   4,  9, 18, 36
   7, 15, 30, 60
   8, 17, 34, 68
while the top left corner of A277820 is:
   1,  3,  5, 15
   2,  6, 10, 30
   7,  9, 27, 45
   4, 12, 20, 60
  13, 23, 57, 75
thus a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 7, a(5) = 6, a(6) = 5, a(7) = 4, a(8) = 13, a(9) = 9, a(12) = 15 and a(15) = 12.

Inverse: A277902.
Cf. A000069, A001969, A010060, A048724, A065621, A115384, A129771, A277823, A277825.
Related permutations and arrays: A277820, A277880, A277881.

If A010060(n) = 1 [when n is one of the odious numbers, A000069], then a(n) = A065621(A115384(n)), otherwise a(n) = A048724(a(floor(n/2))).
As a composition of other permutations:
a(n) = A277820(A277881(n)).
Other identities. For all n >= 1:
A010060(a(n)) = A010060(n). [Preserves the parity of binary weight.]
a(A000069(n)) = A065621(n).
a(A003945(n)) = A001317(n).
a(A129771(n)) = A277823(n).
a(2*A129771(n)) = A277825(n).

A277199 Transpose of square array A277320.

Original entry on oeis.org

1, 2, 2, 7, 4, 3, 4, 14, 6, 4, 13, 8, 9, 8, 5, 14, 26, 12, 28, 10, 6, 11, 28, 23, 16, 27, 12, 7, 8, 22, 18, 52, 20, 18, 14, 8, 25, 16, 29, 56, 57, 24, 21, 16, 9, 26, 50, 24, 44, 54, 46, 28, 56, 18, 10, 31, 52, 43, 32, 39, 36, 35, 32, 63, 20, 11, 28, 62, 46, 100, 40, 58, 42, 104, 36, 54, 22, 12, 21, 56, 33, 104, 125, 48, 49, 112, 101, 40, 49, 24, 13

Offset: 1

Antti Karttunen, Nov 01 2016

See A277320.

			The top left corner of the array:
   1,  2,  7,  4,  13,  14,  11,   8,  25,  26,  31,  28
   2,  4, 14,  8,  26,  28,  22,  16,  50,  52,  62,  56
   3,  6,  9, 12,  23,  18,  29,  24,  43,  46,  33,  36
   4,  8, 28, 16,  52,  56,  44,  32, 100, 104, 124, 112
   5, 10, 27, 20,  57,  54,  39,  40, 125, 114,  99, 108
   6, 12, 18, 24,  46,  36,  58,  48,  86,  92,  66,  72
   7, 14, 21, 28,  35,  42,  49,  56,  79,  70,  93,  84
   8, 16, 56, 32, 104, 112,  88,  64, 200, 208, 248, 224
   9, 18, 63, 36, 101, 126,  83,  72, 209, 202, 231, 252
  10, 20, 54, 40, 114, 108,  78,  80, 250, 228, 198, 216
  11, 22, 49, 44, 127,  98,  69,  88, 227, 254, 217, 196
  12, 24, 36, 48,  92,  72, 116,  96, 172, 184, 132, 144

Transpose: A277320.
Main diagonal: A277699.
Row 1: A065621, Row 3: 3: A277823.
Column 1: A000027, Column 3: A048727.
Cf. A048720.

(define (A277199 n) (A277320bi (A004736 n) (A002260 n))) ;; Code for A277320bi given in A277320.

A(r,c) = A048720(r, A065621(c)).

cp's OEIS Frontend

A129771 Evil odd numbers.

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A277320 Square array A(r,c) = A048720(A065621(r), c), read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

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A277820 Square array: A(r,1) = A065621(r); for c > 1, A(r,c) = A048724(A(r,c-1)), read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

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A277825 a(n) = A048725(A065621(n)) = A048720(A065621(n),5).

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A277902 If A010060(n) = 1, a(n) = A000069(A268671(n)), otherwise a(n) = A001969(1+a(A006068(n)/2)).

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A277819 Transpose of square array A277820.

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A277901 If A010060(n) = 1, a(n) = A065621(A115384(n)), otherwise a(n) = A048724(a(floor(n/2))).

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A277199 Transpose of square array A277320.

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