cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A277822 a(n) = index of the column where n is located in array A277880.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Nov 03 2016

Keywords

Comments

Ordinal transform of A277813.
a(n) = 1 + the number of iterations of map k -> A003188(A006068(k)/2) that are required (when starting from k = n) until k is an odious number.

Links

Crossrefs

One more than A277808.
Cf. A000069, A001969, A001511, A003188, A006068, A010059, A010060, A245710, A277813, A277818, A277880.

Formula

a(0) = 0, for n >= 1, a(n) = 1 + (A010059(n)*A001511(n)).
a(0) = 0, for n >= 1, if A010060(n) = 1 [when n is one of the odious numbers, A000069], then a(n) = 1, otherwise a(n) = 1 + a(A003188(A006068(n)/2)).
Other identities. For all n >= 1:
a(n) = 1 + a(floor(n/2)) when A010060(n) = 0.
a(n) = 1+A277808(n).

A277813 a(n) = A115384(A277812(n)) = index of the row where n is located in array A277880.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 5, 3, 2, 6, 1, 7, 8, 4, 9, 5, 3, 10, 2, 11, 12, 6, 1, 13, 14, 7, 15, 8, 4, 16, 17, 9, 5, 18, 3, 19, 20, 10, 2, 21, 22, 11, 23, 12, 6, 24, 1, 25, 26, 13, 27, 14, 7, 28, 29, 15, 8, 30, 4, 31, 32, 16, 33, 17, 9, 34, 5, 35, 36, 18, 3, 37, 38, 19, 39, 20, 10, 40, 2, 41, 42, 21, 43, 22, 11, 44, 45, 23, 12, 46, 6, 47, 48, 24, 1, 49, 50
Offset: 1

Views

Author

Antti Karttunen, Nov 03 2016

Keywords

Comments

a(n) = index of the first odious number which is encountered when we start from k = n and iterate the map k -> A003188(A006068(k)/2).
Ordinal transform of A277822.
This works as a left inverse of A000069 like also does A115384.

Links

Crossrefs

Cf. A000069, A001969, A003188, A006068, A010060, A115384, A268671, A277812, A277880, A277822.
Cf. A003945 (the positions of 1's in this sequence).

Programs

Formula

a(n) = A115384(A277812(n)).
Other identities. For all n >= 1:
a(A000069(n)) = n.
a(n) = a(floor(n/2)) when A010060(n) = 0.

A277881 Inverse permutation to A277880.

Original entry on oeis.org

1, 3, 2, 6, 5, 4, 10, 15, 9, 8, 21, 7, 28, 36, 14, 45, 20, 13, 55, 12, 66, 78, 27, 11, 91, 105, 35, 120, 44, 19, 136, 153, 54, 26, 171, 18, 190, 210, 65, 17, 231, 253, 77, 276, 90, 34, 300, 16, 325, 351, 104, 378, 119, 43, 406, 435, 135, 53, 465, 25, 496, 528, 152, 561, 170, 64, 595, 33, 630, 666, 189, 24, 703, 741, 209, 780, 230, 76, 820, 23
Offset: 1

Views

Author

Antti Karttunen, Nov 03 2016

Keywords

Links

Crossrefs

Inverse: A277880.
Cf. A277813, A277822.
Cf. A028401 (terms at powers of 2).

Programs

  • Scheme
    (define (A277881 n) (let ((row (A277813 n)) (col (A277822 n))) (* (/ 1 2) (- (expt (+ row col) 2) row col col col -2))))

Formula

a(n) = (1/2) * ((c+r)^2 - r - 3*c + 2), where c = A277822(n), and r = A277813(n).

A129771 Evil odd numbers.

Original entry on oeis.org

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

Views

Author

Tanya Khovanova, May 16 2007

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • Mathematica
    Select[Range[300], OddQ[ # ] && EvenQ[DigitCount[ #, 2, 1]] &] (* Stefan Steinerberger, May 17 2007 *)
Select[Range[300], EvenQ[Plus @@ IntegerDigits[ #, 2]] && OddQ[ # ] &]
  • PARI
    is(n)=n%2 && hammingweight(n)%2==0 \\ Charles R Greathouse IV, Mar 21 2013
  • PARI
    a(n)=4*n-if(hammingweight(n-1)%2,3,1) \\ Charles R Greathouse IV, Mar 21 2013
  • Python
    def A129771(n): return (((m:=n-1)<<1)+(m.bit_count()&1^1)<<1)+1 # Chai Wah Wu, Mar 09 2023

Formula

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

Extensions

More terms from Stefan Steinerberger, May 17 2007

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

Views

Author

Antti Karttunen, Nov 01 2016

Keywords

Comments

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).

Examples

			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

Links

Crossrefs

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.

Programs

Formula

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)).

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

Views

Author

Antti Karttunen, Nov 03 2016

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Formula

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).

A277882 Dispersion of evil numbers, array transposed.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 8, 15, 18, 20, 24, 11, 17, 30, 36, 40, 48, 13, 23, 34, 60, 72, 80, 96, 14, 27, 46, 68, 120, 144, 160, 192, 16, 29, 54, 92, 136, 240, 288, 320, 384, 19, 33, 58, 108, 184, 272, 480, 576, 640, 768, 21, 39, 66, 116, 216, 368, 544, 960, 1152, 1280, 1536, 22, 43, 78, 132, 232, 432, 736, 1088, 1920, 2304, 2560, 3072
Offset: 1

Views

Author

Antti Karttunen, Nov 03 2016

Keywords

Comments

See comments in A277880.

Examples

			The top left 11 x 11 corner of the array:
     1,    2,    4,    7,    8,    11,    13,    14,    16,    19,    21
     3,    5,    9,   15,   17,    23,    27,    29,    33,    39,    43
     6,   10,   18,   30,   34,    46,    54,    58,    66,    78,    86
    12,   20,   36,   60,   68,    92,   108,   116,   132,   156,   172
    24,   40,   72,  120,  136,   184,   216,   232,   264,   312,   344
    48,   80,  144,  240,  272,   368,   432,   464,   528,   624,   688
    96,  160,  288,  480,  544,   736,   864,   928,  1056,  1248,  1376
   192,  320,  576,  960, 1088,  1472,  1728,  1856,  2112,  2496,  2752
   384,  640, 1152, 1920, 2176,  2944,  3456,  3712,  4224,  4992,  5504
   768, 1280, 2304, 3840, 4352,  5888,  6912,  7424,  8448,  9984, 11008
  1536, 2560, 4608, 7680, 8704, 11776, 13824, 14848, 16896, 19968, 22016

Links

Crossrefs

Transpose: A277880.
Inverse: A277883.
Topmost row: A000069, second topmost: A129771.
Leftmost column: A003945.

Programs

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

Views

Author

Antti Karttunen, Nov 03 2016

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Formula

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).
Showing 1-8 of 8 results.

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