A276949 -id:A276949 - 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)).

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.

A276951 Index of column where n is located in array A276953 (equally: row in A276955).

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 1, 5, 2, 6, 7, 8, 3, 9, 4, 10, 11, 12, 13, 14, 15, 16, 17, 18, 1, 19, 5, 20, 21, 22, 2, 23, 6, 24, 25, 26, 7, 27, 8, 28, 29, 30, 31, 32, 33, 34, 35, 36, 3, 37, 9, 38, 39, 40, 4, 41, 10, 42, 43, 44, 11, 45, 12, 46, 47, 48, 49, 50, 51, 52, 53, 54, 13, 55, 14, 56, 57, 58, 15, 59, 16, 60, 61, 62, 17, 63, 18, 64

Offset: 0

Antti Karttunen, Sep 22 2016

a(0) = 0 by convention, because 0 is not present in arrays A276953 and A276955.

Antti Karttunen, Table of n, a(n) for n = 0..5040
Index entries for sequences related to factorial base representation

Cf. A260736, A266193, A273670, A276950, A276952, A273663.
Cf. arrays A276953 & A276955. A276949 gives their other index.
Cf. also A257681, A275847.

(definec (A276951 n) (cond ((zero? n) n) ((not (zero? (A276950 n))) (A276952 n)) (else (A276951 (A266193 n)))))

a(0) = 0; for n >= 1, if A260736(n) > 0 [when A276950(n) is not zero, when n is in A273670], then a(n) = A276952(n) = 1 + A273663(n), otherwise a(n) = a(A266193(n)).
Other identities. For all n >= 0:
a(n) = A257681(A275847(n)).

A276932 Row 2 of A276953: a(n) = A153880(A273670(n)).

Original entry on oeis.org

2, 8, 12, 14, 26, 32, 36, 38, 50, 56, 60, 62, 72, 74, 78, 80, 84, 86, 122, 128, 132, 134, 146, 152, 156, 158, 170, 176, 180, 182, 192, 194, 198, 200, 204, 206, 242, 248, 252, 254, 266, 272, 276, 278, 290, 296, 300, 302, 312, 314, 318, 320, 324, 326, 362, 368, 372, 374, 386, 392, 396, 398, 410, 416, 420

Offset: 0

Antti Karttunen, Sep 22 2016

Numbers k for which A276949(k) = 2. Starting offset is 0 (with a(0) = 2) to match with the starting offset of A273670.

Antti Karttunen, Table of n, a(n) for n = 0..10000
Index entries for sequences related to factorial base representation

Cf. A007623, A153880, A273670.
Row 2 of A276953, column 2 of A276955, positions of 2's in A276949.
Cf. also A276931 (terms halved).

a(n) = A153880(A273670(n)).

A276933 Row 3 of A276953: a(n) = A153880(A153880(A273670(n))).

Original entry on oeis.org

6, 30, 48, 54, 126, 150, 168, 174, 246, 270, 288, 294, 360, 366, 384, 390, 408, 414, 726, 750, 768, 774, 846, 870, 888, 894, 966, 990, 1008, 1014, 1080, 1086, 1104, 1110, 1128, 1134, 1446, 1470, 1488, 1494, 1566, 1590, 1608, 1614, 1686, 1710, 1728, 1734, 1800, 1806, 1824, 1830, 1848, 1854, 2166, 2190, 2208, 2214, 2286, 2310

Offset: 0

Antti Karttunen, Sep 23 2016

All terms are multiples of 6. See A276934.

Antti Karttunen, Table of n, a(n) for n = 0..10000

Row 3 of A276953, column 3 of A276955.
Cf. A153880, A273670, A276932, A276934 (terms divided by six).
Indices of 3's in A276949.

(define (A276933 n) (A153880 (A153880 (A273670 n))))

a(n) = A153880(A153880(A273670(n))) = A153880(A276932(n)).

a(n) = 6*A276934(n).

A276954 Inverse permutation to A276953.

Original entry on oeis.org

1, 3, 2, 4, 7, 6, 11, 5, 16, 22, 29, 8, 37, 12, 46, 56, 67, 79, 92, 106, 121, 137, 154, 10, 172, 17, 191, 211, 232, 9, 254, 23, 277, 301, 326, 30, 352, 38, 379, 407, 436, 466, 497, 529, 562, 596, 631, 13, 667, 47, 704, 742, 781, 18, 821, 57, 862, 904, 947, 68, 991, 80, 1036, 1082, 1129, 1177, 1226, 1276, 1327, 1379, 1432, 93

Offset: 1

Antti Karttunen, Sep 22 2016

Inverse: A276953.
Cf. A276949, A276951.
Related or similar permutations: A257504, A275847.

(define (A276954 n) (let ((col (A276951 n)) (row (A276949 n))) (* (/ 1 2) (- (expt (+ row col) 2) row col col col -2))))

a(n) = (1/2) * ((c+r)^2 - r - 3*c + 2), where c = A276951(n), and r = A276949(n).
As a composition of other permutations:
a(n) = A257504(A275847(n)).

A276956 Inverse permutation to A276955.

Original entry on oeis.org

1, 2, 3, 6, 10, 4, 15, 5, 21, 28, 36, 9, 45, 14, 55, 66, 78, 91, 105, 120, 136, 153, 171, 7, 190, 20, 210, 231, 253, 8, 276, 27, 300, 325, 351, 35, 378, 44, 406, 435, 465, 496, 528, 561, 595, 630, 666, 13, 703, 54, 741, 780, 820, 19, 861, 65, 903, 946, 990, 77, 1035, 90, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431, 1485, 104

Offset: 1

Antti Karttunen, Sep 22 2016

Inverse: A276955.
Cf. A276949, A276951.
Related permutations: A257506, A275847.

(define (A276956 n) (let ((row (A276951 n)) (col (A276949 n))) (* (/ 1 2) (- (expt (+ row col) 2) row col col col -2))))

a(n) = (1/2) * ((c+r)^2 - r - 3*c + 2), where c = A276949(n), and r = A276951(n).
As a composition of other permutations:
a(n) = A257506(A275847(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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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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A276951 Index of column where n is located in array A276953 (equally: row in A276955).

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A276932 Row 2 of A276953: a(n) = A153880(A273670(n)).

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A276933 Row 3 of A276953: a(n) = A153880(A153880(A273670(n))).

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A276954 Inverse permutation to A276953.

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A276956 Inverse permutation to A276955.

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