A153880 -id:A153880 - 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)).

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)).

A275959 Sum of distinct terms of A002674: a(0) = 0, a(2n) = A255411(A153880(a(n))), a(2n+1) = 1+A255411(A153880(a(n))).

Original entry on oeis.org

0, 1, 12, 13, 360, 361, 372, 373, 20160, 20161, 20172, 20173, 20520, 20521, 20532, 20533, 1814400, 1814401, 1814412, 1814413, 1814760, 1814761, 1814772, 1814773, 1834560, 1834561, 1834572, 1834573, 1834920, 1834921, 1834932, 1834933, 239500800, 239500801, 239500812, 239500813, 239501160, 239501161, 239501172, 239501173, 239520960, 239520961

Offset: 0

Antti Karttunen, Aug 16 2016

Fixed points of involution A225901.

This can be also viewed as a function that reinterprets base-2 representation of n in base-((2n)!/2) where the digits are multiplied with the successive terms of A002674, thus a(0) = 0.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to factorial base representation

Cf. A002674, A059590, A153880, A255411, A276082, A276083, A276091.
Fixed points of A225901.
Subsequence of A275956 and of A276089.

from sympy import factorial as f
def a007623(n, p=2): return n if n0 else '0' for i in x)[::-1]
    return 0 if n==0 else sum([int(y[i])*f(i + 1) for i in range(len(y))])
def a153880(n):
    x=(str(a007623(n)) + '0')[::-1]
    return 0 if n==0 else sum([int(x[i])*f(i + 1) for i in range(len(x))])
def a(n): return 0 if n==0 else a255411(a153880(a(n//2))) if n%2==0 else 1 + a255411(a153880(a((n - 1)//2)))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 20 2017

a(0) = 0, a(2n) = A255411(A153880(a(n))), a(2n+1) = 1+A255411(A153880(a(n))).

a(n) = A276089(A276091(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)).

A265905 a(1) = 1; for n > 1, a(n) = a(n-1) + A153880(a(n-1)).

Original entry on oeis.org

1, 3, 11, 49, 291, 1979, 15217, 136659, 1349627, 14561425, 174637707, 2254758155, 31206959833, 467925825795, 7453435202483, 125743951819681, 2262941842058883, 42863071603162571, 852618666050008129, 17902734514975521891, 392964858422866610699, 9001537965557375522737, 216015564123360144707139, 5390978540058458090266187

Offset: 1

Antti Karttunen, Dec 20 2015

In factorial base (A007623) these numbers look as:
  1, 11, 121, 2001, 22011, 242121, 3004001, 33044011, 363524121, 4011111001, 44122221011, 485344431121, 5018801043001, , ...
This sequence is obtained by setting a(1) = 1, and then adding to each previous term a(n-1) the same factorial-base representation, but shifted by one factorial digit left. Only when a term does not contain any adjacent nonzero digits, as is the case with a(4) = "2001" or a(7) = "3004001", does the next term a(5) = "22011" (or respectively a(8) = "33044011") show the uncorrupted "double vision pattern". In other cases, for example, when going from a(2) to a(3), "11" to "121", two nonzero digits are summed up and there is possibly also a carry digit propagating to the left.
Note that the sequence is computed in such a way that factorial-base digits larger than 9 are also correctly summed together. That is, the eventual decimal corruption present in sequences like A007623 does not affect the actual values of this sequence. (See the implementation of A153880.)

Antti Karttunen, Table of n, a(n) for n = 1..120
Index entries for sequences related to factorial base representation.

Row 1 of A275950.
Binomial transform of A275955 (when both are considered as offset-0 sequences).
Cf. A084558 (left inverse), A153880.
Cf. A001710, A265906 (first differences), A265907 (variant).

f[n_] := Module[{k = n, m = 2, r, s = {0}}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, AppendTo[s, r]; m++]; FromDigits[Reverse[s], MixedRadix[Reverse@ Range[2, Length[s] + 1]]]]; NestList[f[#] + # &, 1, 23] (* Amiram Eldar, Feb 14 2024 *)

a(1) = 1; for n > 1, a(n) = a(n-1) + A153880(a(n-1)).
Other identities. For all n >= 1:
A084558(a(n)) = n. [The length of the factorial-base representation of the n-th term is always n.]

Comment and the note about binomial transform corrected - Antti Karttunen, Sep 20 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)).

A265906 a(n) = A153880(A265905(n)); also the first differences of A265905.

Original entry on oeis.org

2, 8, 38, 242, 1688, 13238, 121442, 1212968, 13211798, 160076282, 2080120448, 28952201678, 436718865962, 6985509376688, 118290516617198, 2137197890239202, 40600129761103688, 809755594446845558, 17050115848925513762, 375062123907891088808, 8608573107134508912038, 207014026157802769184402, 5174962975935097945559048

Offset: 1

Antti Karttunen, Dec 20 2015

In factorial base (A007623) these numbers are just like those in A265905, but shifted once left, with an extra zero appended:

10, 110, 1210, 20010, 220110, 2421210, 30040010, 330440110, 3635241210, 40111110010, 441222210110, 4853444311210, 50188010430010, , ...

Antti Karttunen, Table of n, a(n) for n = 1..120
Index entries for sequences related to factorial base representation

Row 2 of A275950.
First differences of A265905.
Cf. A007623, A153880.

a(n) = A153880(A265905(n)).

a(n) = A265905(n+1) - A265905(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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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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A275959 Sum of distinct terms of A002674: a(0) = 0, a(2n) = A255411(A153880(a(n))), a(2n+1) = 1+A255411(A153880(a(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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A265905 a(1) = 1; for n > 1, a(n) = a(n-1) + A153880(a(n-1)).

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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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