A255566 -id:A255566 - cp's OEIS frontend

A257503 Square array A(row,col) read by antidiagonals: A(1,col) = A256450(col-1), and for row > 1, A(row,col) = A255411(A(row-1,col)); Dispersion of factorial base shift A255411 (array transposed).

1, 2, 4, 3, 12, 18, 5, 16, 72, 96, 6, 22, 90, 480, 600, 7, 48, 114, 576, 3600, 4320, 8, 52, 360, 696, 4200, 30240, 35280, 9, 60, 378, 2880, 4920, 34560, 282240, 322560, 10, 64, 432, 2976, 25200, 39600, 317520, 2903040, 3265920, 11, 66, 450, 3360, 25800, 241920, 357840, 3225600, 32659200, 36288000, 13, 70, 456, 3456, 28800, 246240, 2540160, 3588480, 35925120, 399168000, 439084800

Offset: 1

Antti Karttunen, Apr 27 2015

The array is read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

The first row (A256450) contains all the numbers which have at least one 1-digit in their factorial base representation (see A007623), after which the successive rows are obtained from the terms on the row immediately above by shifting their factorial representation one left and then incrementing the nonzero digits in that representation with a factorial base shift-operation A255411.

			The top left corner of the array:
     1,     2,     3,     5,      6,      7,      8,      9,     10,     11,     13
     4,    12,    16,    22,     48,     52,     60,     64,     66,     70,     76
    18,    72,    90,   114,    360,    378,    432,    450,    456,    474,    498
    96,   480,   576,   696,   2880,   2976,   3360,   3456,   3480,   3576,   3696
   600,  3600,  4200,  4920,  25200,  25800,  28800,  29400,  29520,  30120,  30840
  4320, 30240, 34560, 39600, 241920, 246240, 272160, 276480, 277200, 281520, 286560
  ...

Transpose: A257505.
Inverse permutation: A257504.
Row index: A257679, Column index: A257681.
Row 1: A256450, Row 2: A257692, Row 3: A257693.
Columns 1-3: A001563, A062119, A130744 (without their initial zero-terms).
Column 4: A213167 (without the initial one).
Column 5: A052571 (without initial zeros).
Cf. A007623, A256450, A255411.
Cf. also permutations A255565 and A255566.
Thematically similar arrays: A083412, A135764, A246278.

(define (A257503 n) (A257503bi (A002260 n) (A004736 n)))
(define (A257503bi row col) (if (= 1 row) (A256450 (- col 1)) (A255411 (A257503bi (- row 1) col))))

A(1,col) = A256450(col-1), and for row > 1, A(row,col) = A255411(A(row-1,col)).

Formula changed because of the changed starting offset of A256450 - Antti Karttunen, May 30 2016

A257505 Square array A(row,col): A(row,1) = A256450(row-1), and for col > 1, A(row,col) = A255411(A(row,col-1)); Dispersion of factorial base shift A255411.

Original entry on oeis.org

1, 4, 2, 18, 12, 3, 96, 72, 16, 5, 600, 480, 90, 22, 6, 4320, 3600, 576, 114, 48, 7, 35280, 30240, 4200, 696, 360, 52, 8, 322560, 282240, 34560, 4920, 2880, 378, 60, 9, 3265920, 2903040, 317520, 39600, 25200, 2976, 432, 64, 10, 36288000, 32659200, 3225600, 357840, 241920, 25800, 3360, 450, 66, 11, 439084800, 399168000, 35925120, 3588480, 2540160, 246240, 28800, 3456, 456, 70, 13

Offset: 1

Antti Karttunen, Apr 27 2015

The array is read by downward antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

In Kimberling's terminology, this array is called the dispersion of sequence A255411 (when started from its first nonzero term, 4). The left column is the complement of that sequence, which is A256450.

			The top left corner of the array:
   1,   4,  18,   96,   600,   4320,   35280,   322560,   3265920
   2,  12,  72,  480,  3600,  30240,  282240,  2903040,  32659200
   3,  16,  90,  576,  4200,  34560,  317520,  3225600,  35925120
   5,  22, 114,  696,  4920,  39600,  357840,  3588480,  39553920
   6,  48, 360, 2880, 25200, 241920, 2540160, 29030400, 359251200
   7,  52, 378, 2976, 25800, 246240, 2575440, 29352960, 362517120
   8,  60, 432, 3360, 28800, 272160, 2822400, 31933440, 391910400
   9,  64, 450, 3456, 29400, 276480, 2857680, 32256000, 395176320
  10,  66, 456, 3480, 29520, 277200, 2862720, 32296320, 395539200
  11,  70, 474, 3576, 30120, 281520, 2898000, 32618880, 398805120
  13,  76, 498, 3696, 30840, 286560, 2938320, 32981760, 402433920
  14,  84, 552, 4080, 33840, 312480, 3185280, 35562240, 431827200
  15,  88, 570, 4176, 34440, 316800, 3220560, 35884800, 435093120
  17,  94, 594, 4296, 35160, 321840, 3260880, 36247680, 438721920
  19, 100, 618, 4416, 35880, 326880, 3301200, 36610560, 442350720
  20, 108, 672, 4800, 38880, 352800, 3548160, 39191040, 471744000
  21, 112, 690, 4896, 39480, 357120, 3583440, 39513600, 475009920
  23, 118, 714, 5016, 40200, 362160, 3623760, 39876480, 478638720
  ...

Antti Karttunen, Table of n, a(n) for n = 1..1275; the first 50 antidiagonals of array
Clark Kimberling, Interspersions and Dispersions.
Clark Kimberling, Interspersions and Dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
Index entries for sequences related to factorial base representation
Index entries for sequences that are permutations of the natural numbers

Transpose: A257503.
Inverse permutation: A257506.
Row index: A257681, Column index: A257679.
Columns 1-3: A256450, A257692, A257693.
Rows 1-3: A001563, A062119, A130744 (without their initial zero-terms).
Row 4: A213167 (without the initial one).
Row 5: A052571 (without initial zeros).
Cf. A007623, A256450, A255411.
Cf. also permutations A255565, A255566.
Thematically similar arrays: A035513, A054582, A246279.

(define (A257505 n) (A257505bi (A002260 n) (A004736 n)))
(define (A257505bi row col) (if (= 1 col) (A256450 (- row 1)) (A255411 (A257505bi row (- col 1)))))

A(row,1) = A256450(row-1), and for col > 1, A(row,col) = A255411(A(row,col-1)).

Formula changed because of the changed starting offset of A256450 - Antti Karttunen, May 30 2016

A273667 Permutation of nonnegative integers: a(0) = 0, a(A153880(n)) = A255411(a(n)), a(A273670(n)) = A256450(a(n)).

Original entry on oeis.org

0, 1, 4, 2, 6, 3, 18, 8, 12, 5, 24, 10, 48, 15, 16, 7, 30, 13, 56, 20, 21, 9, 36, 17, 96, 67, 60, 26, 27, 11, 72, 42, 22, 23, 120, 81, 240, 73, 66, 32, 33, 14, 87, 49, 28, 29, 144, 101, 360, 270, 88, 89, 80, 38, 90, 39, 52, 19, 107, 57, 288, 34, 76, 35, 168, 125, 416, 303, 109, 110, 99, 44, 420, 111, 108, 45, 61, 25, 112, 131, 64, 68, 327, 40

Offset: 0

Antti Karttunen, May 30 2016

Antti Karttunen, Table of n, a(n) for n = 0..5039
Indranil Ghosh, Python program for computing this sequence
Index entries for sequences related to factorial base representation
Index entries for sequences that are permutations of the natural numbers

Inverse: A273668.
Cf. A153880, A225901, A255411, A256450, A257680, A266193, A273663, A273670.
Similar or related permutations: A255566, A273665.

a(0) = 0; for n >= 1: if A257680(A225901(n)) = 0 [when n is one of the terms of A153880] then a(n) = A255411(a(A266193(n))), otherwise [when n is one of the terms of A273670], a(n) = A256450(a(A273663(n))).
As a composition of other permutations:
a(n) = A255566(A273665(n)).

A255565 a(0) = 0; for n >= 1: if n = A255411(k) for some k, then a(n) = 2a(k), otherwise, n = A256450(h) for some h, and a(n) = 1 + 2a(h).

Original entry on oeis.org

0, 1, 3, 7, 2, 15, 5, 31, 11, 63, 23, 127, 6, 47, 255, 13, 14, 95, 4, 511, 27, 29, 30, 191, 9, 1023, 55, 59, 61, 383, 19, 2047, 111, 119, 123, 767, 39, 4095, 223, 239, 247, 1535, 79, 8191, 447, 479, 495, 3071, 10, 159, 16383, 895, 62, 959, 991, 6143, 21, 319, 32767, 1791, 22, 125, 1919, 1983, 126, 12287, 46, 43, 639, 65535, 254, 3583, 12

Offset: 0

Antti Karttunen, May 05 2015

Because all terms of A255411 are even it means that even terms can only occur in even positions (together with some odd terms, for each one of which there is a separate infinite cycle), while terms in odd positions are all odd.

Inverse: A255566.
Cf. A000079, A000142, A001511, A001563, A083318, A255411, A256450, A257679, A257680, A257682, A257685.
Cf. also arrays A257503, A257505.
Related or similar permutations: A273665, A273668.

a(0) = 0; for n >= 1: if A257680(n) = 0 [i.e., n is one of the terms of A255411], then a(n) = 2*a(A257685(n)), otherwise [when n is one of the terms of A256450], a(n) = 1 + 2*a(A273662(n)).
Other identities:
For all n >= 1, A001511(a(n)) = A257679(n).
For all n >= 1, a(A001563(n)) = A000079(n-1) = 2^(n-1).
For all n >= 1, a(A000142(n)) = A083318(n-1).

Formula changed because of the changed starting offset of A256450 - Antti Karttunen, May 30 2016

A273666 Permutation of nonnegative integers: a(0) = 0; after which, a(2n) = A153880(a(n)), a(2n+1) = A273670(a(n)).

Original entry on oeis.org

0, 1, 2, 3, 6, 4, 8, 5, 24, 10, 12, 7, 30, 13, 14, 9, 120, 34, 36, 16, 48, 18, 26, 11, 144, 42, 50, 19, 54, 20, 32, 15, 720, 154, 156, 46, 168, 49, 60, 22, 240, 66, 72, 25, 126, 37, 38, 17, 840, 186, 192, 58, 246, 68, 74, 27, 264, 73, 78, 28, 150, 44, 56, 21, 5040, 874, 876, 199, 888, 202, 204, 64, 960, 216, 242, 67, 288, 82, 84, 31

Offset: 0

Antti Karttunen, May 30 2016

This sequence can be represented as a binary tree. Each left hand child is produced as A153880(n), and each right hand child as A273670(n), when their parent contains n >= 1:
                                      0
                                      |
                   ...................1...................
                  2                                       3
        6......../ \........4                   8......../ \........5
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
   24       10         12       7          30       13         14       9
120  34   36  16     48  18   26 11     144  42   50  19     54  20   32 15
etc.

Inverse: A273665.
Cf. A153880, A273670.
Related or similar permutations: A255566, A273668.

a(0) = 0; after which, a(2n) = A153880(a(n)), a(2n+1) = A273670(a(n)).

A255567 a(1) = 1, a(2) = 2, after which, a(2n+1) = 1 + a(2n), a(2n) = A255411(a(n)).

Original entry on oeis.org

1, 2, 3, 12, 13, 16, 17, 72, 73, 76, 77, 90, 91, 94, 95, 480, 481, 484, 485, 498, 499, 502, 503, 576, 577, 580, 581, 594, 595, 598, 599, 3600, 3601, 3604, 3605, 3618, 3619, 3622, 3623, 3696, 3697, 3700, 3701, 3714, 3715, 3718, 3719, 4200, 4201, 4204, 4205, 4218, 4219, 4222, 4223, 4296, 4297, 4300, 4301, 4314, 4315, 4318, 4319

Offset: 1

Antti Karttunen, May 05 2015

nonn

From 2 onward, the sequence seems to give those n for which A256450(A255411(n))+1 = A255411(A256450(n)), i.e., grandparents for those cousins in tree A255566 where the cousin at the right side is one more than the cousin at the left side.

			This sequence can be represented as a binary tree. Apart from the 1 at root, and its children 2 and 3, from then on each left hand child is produced as A255411(n), and each right hand child as 1 + A255411(n) when parent contains n >= 2:
                   ..................1..................
                  2                                     3
        12......./ \.......13                 16......./ \.......17
       / \                 / \               / \                 / \
      /   \               /   \             /   \               /   \
     /     \             /     \           /     \             /     \
   72       73         76       77       90       91         94       95
480  481 484  485   498  499 502  503 576  577 580  581   594  595 598  599
etc.

Antti Karttunen, Table of n, a(n) for n = 1..1023

Cf. A256450, A255411. See also the tree illustration in A255566.

a(1) = 1, a(2) = 2, after which, a(2n+1) = 1 + a(2n), a(2n) = A255411(a(n)).

Edited because of the changed starting offset of A256450 - Antti Karttunen, May 30 2016

cp's OEIS Frontend

A257503 Square array A(row,col) read by antidiagonals: A(1,col) = A256450(col-1), and for row > 1, A(row,col) = A255411(A(row-1,col)); Dispersion of factorial base shift A255411 (array transposed).

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A257505 Square array A(row,col): A(row,1) = A256450(row-1), and for col > 1, A(row,col) = A255411(A(row,col-1)); Dispersion of factorial base shift A255411.

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A273667 Permutation of nonnegative integers: a(0) = 0, a(A153880(n)) = A255411(a(n)), a(A273670(n)) = A256450(a(n)).

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A255565 a(0) = 0; for n >= 1: if n = A255411(k) for some k, then a(n) = 2*a(k), otherwise, n = A256450(h) for some h, and a(n) = 1 + 2*a(h).

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A273666 Permutation of nonnegative integers: a(0) = 0; after which, a(2n) = A153880(a(n)), a(2n+1) = A273670(a(n)).

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Formula

A255567 a(1) = 1, a(2) = 2, after which, a(2n+1) = 1 + a(2n), a(2n) = A255411(a(n)).

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A255565 a(0) = 0; for n >= 1: if n = A255411(k) for some k, then a(n) = 2a(k), otherwise, n = A256450(h) for some h, and a(n) = 1 + 2a(h).