A276953 -id:A276953 - cp's OEIS frontend

A276949 Index of row where n is located in array A276953 (equally: column in A276955).

0, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5

Offset: 0

Antti Karttunen, Sep 22 2016

This is the smallest difference that occurs between any nonzero digit's radix (which is one more than its one-based position from the right) and that digit's value in the factorial base representation of n. See A225901 and the example.

a(0) = 0 by convention, as there are no nonzero digits present, and neither does 0 occur in arrays A276953 & A276955.

			For n=8, its factorial base representation (A007623) is "110", where the radix for each digit position 1, 2, 3 (from the right) is 2, 3, 4 (one larger than the position). For the 1 in the middle position the difference is 3-1 = 2, while for the 1 at the left we obtain 4-1 = 3. Of these two differences 2 is smaller, thus a(8)=2.

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

Cf. A007623, A257679, A266193, A276950.
Cf. A276951 (for the other index).
Cf. arrays A276953 & A276955.
Cf. also A225901, A273667, A275847.

a(0) = 0, and for n >= 1: if A276950(n) = 1, then a(n) = 1, otherwise a(n) = 1 + a(A266193(n)).
Other identities. For all n >= 0:
a(n) = A257679(A225901(n)) = A257679(A275847(n)) = A257679(A273667(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)).

A276616 Square array A(n,k) = A276953(n,k)/n!, 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, 1, 4, 4, 1, 5, 6, 5, 1, 7, 7, 8, 6, 1, 9, 13, 9, 10, 7, 1, 10, 16, 21, 11, 12, 8, 1, 11, 18, 25, 31, 13, 14, 9, 1, 13, 19, 28, 36, 43, 15, 16, 10, 1, 15, 25, 29, 40, 49, 57, 17, 18, 11, 1, 16, 28, 41, 41, 54, 64, 73, 19, 20, 12, 1, 17, 30, 45, 61, 55, 70, 81, 91, 21, 22, 13, 1, 18, 31, 48, 66, 85, 71, 88, 100, 111, 23, 24, 14, 1

Offset: 1

Antti Karttunen, Sep 22 2016

			The top left corner of the array:
  1, 3,  4,  5,  7,  9, 10, 11,  13,  15,  16,  17,  18,  19,  20,  21,  22
  1, 4,  6,  7, 13, 16, 18, 19,  25,  28,  30,  31,  36,  37,  39,  40,  42
  1, 5,  8,  9, 21, 25, 28, 29,  41,  45,  48,  49,  60,  61,  64,  65,  68
  1, 6, 10, 11, 31, 36, 40, 41,  61,  66,  70,  71,  90,  91,  95,  96, 100
  1, 7, 12, 13, 43, 49, 54, 55,  85,  91,  96,  97, 126, 127, 132, 133, 138
  1, 8, 14, 15, 57, 64, 70, 71, 113, 120, 126, 127, 168, 169, 175, 176, 182

Transpose: A276617.
Cf. A000142, A276953.
Row 1: A273670, row 2: A276931, row 3: A276934.
For columns, see the rows of transpose A276617.

(define (A276616 n) (A276616bi (A002260 n) (A004736 n)))
(define (A276616bi row col) (/ (A276953bi row col) (A000142 row)))

A(n,k) = A276953(n,k)/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)).

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.

Original entry on oeis.org

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)).

A276943 Square array A(row,col) read by antidiagonals: A(1,col) = A276155(col), and for row > 1, A(row,col) = A276154(A(row-1,col)); Dispersion of primorial base left shift A276154 (array transposed).

Original entry on oeis.org

1, 3, 2, 4, 8, 6, 5, 12, 36, 30, 7, 14, 60, 240, 210, 9, 32, 66, 420, 2520, 2310, 10, 38, 216, 450, 4620, 32340, 30030, 11, 42, 246, 2340, 4830, 60060, 540540, 510510, 13, 44, 270, 2550, 30240, 62370, 1021020, 10210200, 9699690, 15, 62, 276, 2730, 32550, 512820, 1051050, 19399380, 232792560, 223092870

Offset: 1

Antti Karttunen, Sep 24 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 A002110(n-1).

			The top left corner of the array:
    1,    3,    4,    5,     7,     9,    10,    11,    13,    15,    16
    2,    8,   12,   14,    32,    38,    42,    44,    62,    68,    72
    6,   36,   60,   66,   216,   246,   270,   276,   426,   456,   480
   30,  240,  420,  450,  2340,  2550,  2730,  2760,  4650,  4860,  5040
  210, 2520, 4620, 4830, 30240, 32550, 34650, 34860, 60270, 62580, 64680

Inverse permutation: A276944.
Transpose: A276945.
Column 1: A002110, Row 1: A276155.
Cf. A276154.
Cf. also array A276953.

(define (A276943 n) (A276943bi (A002260 n) (A004736 n)))
(define (A276943bi row col) (if (= 1 row) (A276155 col) (A276154 (A276943bi (- row 1) col))))

A(1,col) = A276155(col); for row > 1, A(row,col) = A276154(A(row-1,col)).

A276941 Square array A(row,col): A(1,col) = A276937(col), and for row > 1, A(row,col) = A003961(A(row-1,col)), read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

2, 6, 3, 9, 15, 5, 10, 25, 35, 7, 14, 21, 49, 77, 11, 18, 33, 55, 121, 143, 13, 22, 75, 65, 91, 169, 221, 17, 26, 39, 245, 119, 187, 289, 323, 19, 30, 51, 85, 847, 209, 247, 361, 437, 23, 34, 105, 95, 133, 1859, 299, 391, 529, 667, 29, 38, 57, 385, 161, 253, 3757, 493, 551, 841, 899, 31, 42, 69, 115, 1001, 319, 377, 6137, 589, 713, 961, 1147, 37

Offset: 2

Antti Karttunen, Sep 25 2016

The starting offset is 2 because 1 is not included in the array proper. With it the terms are a permutation of A276078.

			The top left corner of the array:
   2,   6,   9,   10,   14,    18,   22,   26,    30,   34,   38,    42
   3,  15,  25,   21,   33,    75,   39,   51,   105,   57,   69,   165
   5,  35,  49,   55,   65,   245,   85,   95,   385,  115,  145,   455
   7,  77, 121,   91,  119,   847,  133,  161,  1001,  203,  217,  1309
  11, 143, 169,  187,  209,  1859,  253,  319,  2431,  341,  407,  2717
  13, 221, 289,  247,  299,  3757,  377,  403,  4199,  481,  533,  5083
  17, 323, 361,  391,  493,  6137,  527,  629,  7429,  697,  731,  9367
  19, 437, 529,  551,  589, 10051,  703,  779, 12673,  817,  893, 13547
  23, 667, 841,  713,  851, 19343,  943,  989, 20677, 1081, 1219, 24679
  29, 899, 961, 1073, 1189, 27869, 1247, 1363, 33263, 1537, 1711, 36859

Antti Karttunen, Table of n, a(n) for n = 2..1276; the first 50 antidiagonals of array

Transpose: A276942.
Topmost row: A276937, second row: A276938. Leftmost column: A000040.
Cf. A003961.
Cf. A276078 (sorted into ascending order).
Cf. also A276075, A276953.

(define (A276941 n) (A276941bi (A002260 (- n 1)) (A004736 (- n 1))))
(define (A276941bi row col) (if (= 1 row) (A276937 col) (A003961 (A276941bi (- row 1) col))))

A(1,col) = A276937(col), and for row > 1, A(row,col) = A003961(A(row-1,col)).

cp's OEIS Frontend

A276949 Index of row where n is located in array A276953 (equally: column in A276955).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A276616 Square array A(n,k) = A276953(n,k)/n!, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Scheme

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A276954 Inverse permutation to A276953.

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Scheme

Formula

A276943 Square array A(row,col) read by antidiagonals: A(1,col) = A276155(col), and for row > 1, A(row,col) = A276154(A(row-1,col)); Dispersion of primorial base left shift A276154 (array transposed).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Scheme

Formula

A276941 Square array A(row,col): A(1,col) = A276937(col), and for row > 1, A(row,col) = A003961(A(row-1,col)), read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.