A273670 -id:A273670 - cp's OEIS frontend

A276955 Square array A(row,col): A(row,1) = A273670(row-1), and for col > 1, A(row,col) = A153880(A(row,col-1)); Dispersion of factorial base left shift A153880.

1, 2, 3, 6, 8, 4, 24, 30, 12, 5, 120, 144, 48, 14, 7, 720, 840, 240, 54, 26, 9, 5040, 5760, 1440, 264, 126, 32, 10, 40320, 45360, 10080, 1560, 744, 150, 36, 11, 362880, 403200, 80640, 10800, 5160, 864, 168, 38, 13, 3628800, 3991680, 725760, 85680, 41040, 5880, 960, 174, 50, 15, 39916800, 43545600, 7257600, 766080, 367920, 46080, 6480, 984, 246, 56, 16

Offset: 1

Antti Karttunen, Sep 22 2016

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

When viewed in factorial base (A007623) the terms on each row start all with the same prefix, but with an increasing number of zeros appended to the end. For example, for row 8 (A001344 from a(1)=11 onward), the terms written in factorial base look as: 121, 1210, 12100, 121000, ...

			The top left {1..9} x {1..18} corner of the array:
   1,  2,   6,   24,   120,    720,    5040,    40320,    362880
   3,  8,  30,  144,   840,   5760,   45360,   403200,   3991680
   4, 12,  48,  240,  1440,  10080,   80640,   725760,   7257600
   5, 14,  54,  264,  1560,  10800,   85680,   766080,   7620480
   7, 26, 126,  744,  5160,  41040,  367920,  3669120,  40279680
   9, 32, 150,  864,  5880,  46080,  408240,  4032000,  43908480
  10, 36, 168,  960,  6480,  50400,  443520,  4354560,  47174400
  11, 38, 174,  984,  6600,  51120,  448560,  4394880,  47537280
  13, 50, 246, 1464, 10200,  81360,  730800,  7297920,  80196480
  15, 56, 270, 1584, 10920,  86400,  771120,  7660800,  83825280
  16, 60, 288, 1680, 11520,  90720,  806400,  7983360,  87091200
  17, 62, 294, 1704, 11640,  91440,  811440,  8023680,  87454080
  18, 72, 360, 2160, 15120, 120960, 1088640, 10886400, 119750400
  19, 74, 366, 2184, 15240, 121680, 1093680, 10926720, 120113280
  20, 78, 384, 2280, 15840, 126000, 1128960, 11249280, 123379200
  21, 80, 390, 2304, 15960, 126720, 1134000, 11289600, 123742080
  22, 84, 408, 2400, 16560, 131040, 1169280, 11612160, 127008000
  23, 86, 414, 2424, 16680, 131760, 1174320, 11652480, 127370880

Inverse permutation: A276956.
Transpose: A276953.
Cf. A276949 (index of column where n appears), A276951 (index of row).
Cf. A153880.
Columns 1-3: A273670, A276932, A276933.
The following lists some of the rows that have their own entries. Pattern present in the factorial base expansion of the terms on that row is given in double quotes:
Row 1: A000142 (from a(1)=1, "1" onward),
Row 2: A001048 (from a(2)=3, "11" onward),
Row 3: A052849 (from a(2)=4, "20" onward).
Row 4: A052649 (from a(1)=5, "21" onward).
Row 5: A108217 (from a(3)=7, "101" onward).
Row 6: A054119 (from a(3)=9, "111" onward).
Row 7: A052572 (from a(2)=10, "120" onward).
Row 8: A001344 (from a(1)=11, "121" onward).
Row 13: A052560 (from a(3)=18, "300" onward).
Row 16: A225658 (from a(1)=21, "311" onward).
Row 20: A276940 (from a(3) = 27, "1011" onward).
Related or similar permutations: A257505, A275848, A273666.
Cf. also arrays A276617, A276588 & A276945.

(define (A276955 n) (A276955bi (A002260 n) (A004736 n)))
(define (A276955bi row col) (if (= 1 col) (A273670 (- row 1)) (A153880 (A276955bi row (- col 1)))))

A(row,1) = A273670(row-1), and for col > 1, A(row,col) = A153880(A(row,col-1))
As a composition of other permutations:
a(n) = A275848(A257505(n)).

A273663 Least monotonic left inverse for A273670: a(1) = 0; for n > 1, a(n) = A257680(A225901(n)) + a(n-1).

Original entry on oeis.org

0, 0, 1, 2, 3, 3, 4, 4, 5, 6, 7, 7, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 17, 18, 18, 19, 20, 21, 21, 22, 22, 23, 24, 25, 25, 26, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 35, 36, 36, 37, 38, 39, 39, 40, 40, 41, 42, 43, 43, 44, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 53, 54, 54, 55, 56, 57, 57, 58, 58, 59, 60, 61, 61

Offset: 1

Antti Karttunen, May 30 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10080

Left inverse of A273670.
Cf. A225901, A257680.
Cf. also A273662.

from sympy import factorial as f
def a007623(n, p=2): return n if nIndranil Ghosh, Jun 24 2017

a(1) = 0; for n > 1, a(n) = A257680(A225901(n)) + a(n-1).
Other identities.
For all n >= 0, a(A273670(n)) = n.

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)).

A276953 Square array A(row,col) read by antidiagonals: A(1,col) = A273670(col-1), and for row > 1, A(row,col) = A153880(A(row-1,col)); Dispersion of factorial base shift A153880 (array transposed).

Original entry on oeis.org

1, 3, 2, 4, 8, 6, 5, 12, 30, 24, 7, 14, 48, 144, 120, 9, 26, 54, 240, 840, 720, 10, 32, 126, 264, 1440, 5760, 5040, 11, 36, 150, 744, 1560, 10080, 45360, 40320, 13, 38, 168, 864, 5160, 10800, 80640, 403200, 362880, 15, 50, 174, 960, 5880, 41040, 85680, 725760, 3991680, 3628800, 16, 56, 246, 984, 6480, 46080, 367920, 766080, 7257600, 43545600, 39916800

Offset: 1

Antti Karttunen, Sep 22 2016

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

Entries on row n are all multiples of n!. Dividing that factor out gives another array A276616.

			The top left corner of the array:
    1,    3,     4,     5,     7,     9,    10,    11,    13,    15,    16
    2,    8,    12,    14,    26,    32,    36,    38,    50,    56,    60
    6,   30,    48,    54,   126,   150,   168,   174,   246,   270,   288
   24,  144,   240,   264,   744,   864,   960,   984,  1464,  1584,  1680
  120,  840,  1440,  1560,  5160,  5880,  6480,  6600, 10200, 10920, 11520
  720, 5760, 10080, 10800, 41040, 46080, 50400, 51120, 81360, 86400, 90720

Inverse permutation: A276954.
Transpose: A276955.
Cf. A276949 (index of row where n appears), A276951 (index of column).
Cf. A153880, A273670.
Row 1: A273670, Row 2: A276932, Row 3: A276933.
Column 1: A000142. For other columns, see the rows of transposed array A276955.
Related or similar permutations: A257503, A275848, A273666.
Cf. also arrays A276616, A276589 & A276943.

(define (A276953 n) (A276953bi (A002260 n) (A004736 n)))
(define (A276953bi row col) (if (= 1 row) (A273670 (- col 1)) (A153880 (A276953bi (- row 1) col))))

A(1,col) = A273670(col-1), and for row > 1, A(row,col) = A153880(A(row-1,col))
As a composition of other permutations:
a(n) = A275848(A257503(n)).
Other identities. For all n >= 1:
A(A276949(n),A276951(n)) = n.

A273668 Permutation of nonnegative integers: a(0) = 0, a(A255411(n)) = A153880(a(n)), a(A256450(n)) = A273670(a(n)).

Original entry on oeis.org

0, 1, 3, 5, 2, 9, 4, 15, 7, 21, 11, 29, 8, 17, 41, 13, 14, 23, 6, 57, 19, 20, 32, 33, 10, 77, 27, 28, 44, 45, 16, 101, 39, 40, 61, 63, 22, 129, 53, 55, 83, 87, 31, 165, 71, 75, 107, 111, 12, 43, 213, 95, 56, 99, 137, 141, 18, 59, 269, 119, 26, 76, 125, 177, 80, 183, 38, 25, 81, 341, 134, 153, 30, 37, 100, 161, 62, 225, 104, 231, 52, 35

Offset: 0

Antti Karttunen, May 30 2016

Inverse: A273667.
Cf. A153880, A255411, A256450, A257680, A257684, A273662, A273670.
Similar or related permutations: A255565, A273666.

a(0) = 0; for n >= 1: if A257680(n) = 0 [when n is one of the terms of A255411] then a(n) = A153880(a(A257684(n))), otherwise [when n is one of the terms of A256450], a(n) = A273670(a(A273662(n))).
As a composition of other permutations:
a(n) = A273666(A255565(n)).

A275847 Permutation of natural numbers: a(0) = 0, a(A153880(n)) = A255411(a(n)), a(A273670(n)) = A256450(n).

Original entry on oeis.org

0, 1, 4, 2, 3, 5, 18, 6, 12, 7, 8, 9, 16, 10, 22, 11, 13, 14, 15, 17, 19, 20, 21, 23, 96, 24, 48, 25, 26, 27, 72, 28, 52, 29, 30, 31, 60, 32, 64, 33, 34, 35, 36, 37, 38, 39, 40, 41, 90, 42, 66, 43, 44, 45, 114, 46, 70, 47, 49, 50, 76, 51, 84, 53, 54, 55, 56, 57, 58, 59, 61, 62, 88, 63, 94, 65, 67, 68, 100, 69, 108, 71, 73, 74, 112, 75, 118, 77, 78

Offset: 0

Antti Karttunen, Aug 13 2016

Inverse: A275848.
Cf. A153880, A225901, A255411, A256450, A257680, A266193, A273663, A273670.
Similar permutations: A273667 (a more recursed variant), A275845, A275846.

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(A273663(n)).

A275841 Permutation of nonnegative integers: a(n) = A273663(A275837(A273670(n))).

Original entry on oeis.org

0, 2, 12, 3, 16, 1, 72, 14, 11, 13, 90, 8, 70, 6, 7, 17, 84, 9, 58, 76, 77, 15, 78, 5, 480, 53, 36, 94, 95, 10, 47, 54, 74, 75, 576, 19, 474, 45, 52, 88, 89, 4, 33, 71, 92, 93, 552, 483, 358, 449, 26, 28, 21, 82, 29, 83, 62, 73, 501, 60, 431, 86, 43, 87, 528, 579, 206, 417, 493, 496, 485, 57, 203, 497, 492, 55, 38, 91, 494, 597, 48, 50, 294, 80, 288, 24, 25, 81

Offset: 0

Antti Karttunen, Aug 13 2016

Inverse: A275842.

Cf. A273663, A273670, A275837.

(define (A275841 n) (A273663 (A275837 (A273670 n))))

a(n) = A273663(A275837(A273670(n))).

A275842 Permutation of nonnegative integers: a(n) = A273663(A275838(A273670(n))).

Original entry on oeis.org

0, 5, 1, 3, 41, 23, 13, 14, 11, 17, 29, 8, 2, 9, 7, 21, 4, 15, 104, 35, 231, 52, 209, 285, 85, 86, 50, 127, 51, 54, 105, 431, 197, 42, 182, 172, 26, 177, 76, 125, 100, 225, 161, 62, 134, 37, 153, 30, 80, 183, 81, 341, 38, 25, 31, 75, 165, 71, 18, 119, 59, 269, 56, 141, 99, 137, 107, 111, 213, 95, 12, 43, 6, 57, 32, 33, 19, 20, 22, 129, 83, 87, 53, 55, 16, 101

Offset: 0

Antti Karttunen, Aug 13 2016

Inverse: A275841.

Cf. A273663, A273670, A275838.

(define (A275842 n) (A273663 (A275838 (A273670 n))))

a(n) = A273663(A275838(A273670(n))).

A275848 Permutation of natural numbers: a(0) = 0, a(A255411(n)) = A153880(a(n)), a(A256450(n)) = A273670(n).

Original entry on oeis.org

0, 1, 3, 4, 2, 5, 7, 9, 10, 11, 13, 15, 8, 16, 17, 18, 12, 19, 6, 20, 21, 22, 14, 23, 25, 27, 28, 29, 31, 33, 34, 35, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 57, 26, 58, 59, 61, 32, 63, 64, 65, 66, 67, 68, 69, 36, 70, 71, 73, 38, 75, 50, 76, 77, 79, 56, 81, 30, 82, 83, 85, 60, 87, 88, 89, 90, 91, 92, 93, 62, 94, 95, 96, 72, 97, 48

Offset: 0

Antti Karttunen, Aug 13 2016

Inverse: A275847.
Cf. A153880, A255411, A256450, A257680, A257684, A273662, A273670.
Similar permutations: A273668 (a more recursed variant), A275845, A275846.

a(0) = 0; for n >= 1: if A257680(n) = 0 [when n is one of the terms of A255411] then a(n) = A153880(a(A257684(n))), otherwise [when n is one of the terms of A256450], a(n) = A273670(A273662(n)).

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)).

cp's OEIS Frontend

A276955 Square array A(row,col): A(row,1) = A273670(row-1), and for col > 1, A(row,col) = A153880(A(row,col-1)); Dispersion of factorial base left shift A153880.

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A273663 Least monotonic left inverse for A273670: a(1) = 0; for n > 1, a(n) = A257680(A225901(n)) + a(n-1).

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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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A276953 Square array A(row,col) read by antidiagonals: A(1,col) = A273670(col-1), and for row > 1, A(row,col) = A153880(A(row-1,col)); Dispersion of factorial base shift A153880 (array transposed).

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A273668 Permutation of nonnegative integers: a(0) = 0, a(A255411(n)) = A153880(a(n)), a(A256450(n)) = A273670(a(n)).

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A275847 Permutation of natural numbers: a(0) = 0, a(A153880(n)) = A255411(a(n)), a(A273670(n)) = A256450(n).

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A275841 Permutation of nonnegative integers: a(n) = A273663(A275837(A273670(n))).

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A275842 Permutation of nonnegative integers: a(n) = A273663(A275838(A273670(n))).

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A275848 Permutation of natural numbers: a(0) = 0, a(A255411(n)) = A153880(a(n)), a(A256450(n)) = A273670(n).

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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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