A257681 -id:A257681 - 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

A257682 Partial sums of A257680: a(0) = 0; for n >= 1, a(n) = A257680(n) + a(n-1).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 04 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10080

Cf. A257680, A257681.

One more than A273662.

a(0) = 0; for n >= 1, a(n) = A257680(n) + a(n-1).
Other identities:
a(n) = A273662(n)+1.

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

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)).

A257685 Left inverse for injection A255411: a(0) = 0, after which, if n = A255411(k) for some k, then a(n) = k, otherwise a(n) = 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 4, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 9, 0, 10, 0, 0, 0, 11, 0, 12, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 15, 0, 16, 0, 0, 0, 17, 0, 18, 0, 0, 0, 19, 0, 0, 0, 0, 0, 0, 0, 20, 0, 0, 0, 21, 0, 22, 0, 0, 0, 23, 0, 0

Offset: 0

Antti Karttunen, May 04 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10080

Cf. A255411, A257680, A257681, A257684.

Position[Select[Range[0, 120], ! MemberQ[IntegerDigits[#, MixedRadix[Reverse@ Range@ 12]], 1] &], #] - 1 & /@ Range[0, 120] /. {} -> 0 // Flatten (* Michael De Vlieger, May 30 2016, Version 10.2 *)

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==1 else sum([int(y[i])*f(i + 1) for i in range(len(y))])
def a257680(n): return 1 if '1' in str(a007623(n)) else 0
def a(n): return (1 - a257680(n))*a257684(n)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 21 2017

(define (A257685 n) (* (- 1 (A257680 n)) (A257684 n)))

a(0) = 0, after which, if n = A255411(k) for some k, then a(n) = k, otherwise a(n) = 0.
a(n) = (1-A257680(n)) * A257684(n).
Other identities:
For all n >= 0, a(A255411(n)) = n. [This sequence works as a left inverse of A255411.]

A257504 Inverse permutation to A257503.

Original entry on oeis.org

1, 2, 4, 3, 7, 11, 16, 22, 29, 37, 46, 5, 56, 67, 79, 8, 92, 6, 106, 121, 137, 12, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436, 466, 497, 529, 562, 596, 631, 667, 704, 742, 781, 821, 862, 17, 904, 947, 991, 23, 1036, 1082, 1129, 1177, 1226, 1276, 1327, 30, 1379, 1432, 1486, 38, 1541, 47, 1597, 1654, 1712, 57, 1771, 9

Offset: 1

Antti Karttunen, May 06 2015

nonn

Inverse: A257503.

Cf. A257679, A257681.

(define (A257504 n) (let ((col (A257681 n)) (row (A257679 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 = A257681(n), and r = A257679(n).

A257506 Inverse permutation to A257505.

Original entry on oeis.org

1, 3, 6, 2, 10, 15, 21, 28, 36, 45, 55, 5, 66, 78, 91, 9, 105, 4, 120, 136, 153, 14, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 20, 946, 990, 1035, 27, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 35, 1431, 1485, 1540, 44, 1596, 54, 1653, 1711, 1770, 65, 1830, 8

Offset: 1

Antti Karttunen, May 06 2015

nonn

Inverse: A257505.

Cf. A257679, A257681.

(define (A257506 n) (let ((row (A257681 n)) (col (A257679 n))) (* (/ 1 2) (- (expt (+ row col) 2) row col col col -2))))

a(n) = (1/2) * ((c+r)^2 - r - 3*c + 2), where r = A257681(n), and c = A257679(n).

A276947 First differences of A256450: a(n) = A256450(n) - A256450(n-1).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 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, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 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

Offset: 1

Antti Karttunen, Sep 22 2016

Antti Karttunen, Table of n, a(n) for n = 1..5040
Index entries for sequences related to factorial base representation

Cf. A256450.

Cf. also A276948, A257682, A257681.

(define (A276947 n) (- (A256450 n) (A256450 (- n 1))))

a(n) = A256450(n) - A256450(n-1). [Note that the indexing of A256450 begins from zero.]

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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A257682 Partial sums of A257680: a(0) = 0; for n >= 1, a(n) = A257680(n) + a(n-1).

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

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A257685 Left inverse for injection A255411: a(0) = 0, after which, if n = A255411(k) for some k, then a(n) = k, otherwise a(n) = 0.

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A257504 Inverse permutation to A257503.

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A257506 Inverse permutation to A257505.

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A276947 First differences of A256450: a(n) = A256450(n) - A256450(n-1).

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