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 1-6 of 6 results.

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).

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)).

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.

A277818 Index of the column where n is located in array A277820: a(n) = 1 + A268389(n).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Nov 02 2016

Keywords

Comments

Ordinal transform of A268671.

Links

Crossrefs

One more than A268389.
Cf. A277820.
Cf. A268671 for the other index, also A277822.

Programs

Formula

a(n) = 1 + A268389(n).

A268670 a(n) = A006068(A268669(n)).

Original entry on oeis.org

1, 3, 1, 7, 1, 3, 5, 15, 5, 3, 13, 7, 9, 11, 1, 31, 1, 11, 29, 7, 25, 27, 9, 15, 17, 19, 5, 23, 13, 3, 21, 63, 21, 3, 61, 23, 57, 59, 13, 15, 49, 51, 17, 55, 5, 19, 53, 31, 33, 35, 1, 39, 29, 11, 37, 47, 9, 27, 45, 7, 41, 43, 25, 127, 25, 43, 125, 7, 121, 123, 41, 47, 113, 115, 9, 119, 45, 27, 117, 31, 97, 99, 33, 103, 1
Offset: 1

Views

Author

Antti Karttunen, Feb 10 2016

Keywords

Comments

All terms are odd, by definition.

Links

Crossrefs

Cf. A001317 (positions of ones).
Cf. A006068, A268669, A268671.

Programs

Formula

a(n) = A006068(A268669(n)).

A277821 Inverse permutation to A277820.

Original entry on oeis.org

1, 3, 2, 10, 4, 5, 6, 36, 9, 8, 28, 14, 15, 21, 7, 136, 11, 27, 120, 19, 91, 105, 20, 44, 45, 55, 13, 78, 35, 12, 66, 528, 77, 17, 496, 90, 435, 465, 43, 53, 325, 351, 54, 406, 18, 65, 378, 152, 153, 171, 16, 210, 135, 34, 190, 300, 26, 119, 276, 25, 231, 253, 104, 2080, 118, 275, 2016, 32, 1891, 1953, 252, 324, 1653, 1711, 33
Offset: 1

Views

Author

Antti Karttunen, Nov 03 2016

Keywords

Links

Crossrefs

Inverse: A277820.
Cf. A268671, A277818.

Programs

  • Scheme
    (define (A277821 n) (let ((row (A268671 n)) (col (A277818 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 = A277818(n), and r = A268671(n).
Showing 1-6 of 6 results.

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