A257693 -id:A257693 - 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

A257679 The smallest nonzero digit present in the factorial base representation (A007623) of n, 0 if no nonzero digits present.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 04 2015

a(0) = 0 by convention, because "0" has no nonzero digits present.

a(n) gives the row index of n in array A257503 (equally, the column index for array A257505).

			Factorial base representation (A007623) of 4 is "20", the smallest digit which is not zero is "2", thus a(4) = 2.

Antti Karttunen, Table of n, a(n) for n = 0..5040

Positions of records: A001563.
Cf. A256450, A257692, A257693 (positions of 1's, 2's and 3's in this sequence).
Cf. A007623, A051683, A099563, A257680, A257684, A257687.
Cf. also A257079, A246359 and arrays A257503, A257505.

a[n_] := Module[{k = n, m = 2, rmin = n, r}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[0 < r < rmin, rmin = r]; m++]; rmin]; Array[a, 100, 0] (* Amiram Eldar, Jan 23 2024 *)

def A(n, p=2):
    return n if nIndranil Ghosh, Jun 19 2017

(define (A257679 n) (let loop ((n n) (i 2) (mind 0)) (if (zero? n) mind (let ((d (modulo n i))) (loop (/ (- n d) i) (+ 1 i) (cond ((zero? mind) d) ((zero? d) mind) (else (min d mind))))))))
;; Alternative implementations based on given recurrences, using memoizing definec-macro:
(definec (A257679 n) (if (zero? (A257687 n)) (A099563 n) (min (A099563 n) (A257679 (A257687 n)))))
(definec (A257679 n) (cond ((zero? n) n) ((= 1 (A257680 n)) 1) (else (+ 1 (A257679 (A257684 n))))))

If A257687(n) = 0, then a(n) = A099563(n), otherwise a(n) = min(A099563(n), a(A257687(n))).
In other words, if n is either zero or one of the terms of A051683, then a(n) = A099563(n) [the most significant digit of its f.b.r.], otherwise take the minimum of the most significant digit and a(A257687(n)) [value computed by recursing with a smaller value obtained by discarding that most significant digit].
a(0) = 0, and for n >= 1: if A257680(n) = 1, then a(n) = 1, otherwise 1 + a(A257684(n)).
Other identities:
For all n >= 0, a(A001563(n)) = n. [n * n! gives the first position where n appears. Note also that the "digits" (placeholders) in factorial base representation may get arbitrarily large values.]
For all n >= 0, a(2n+1) = 1 [because all odd numbers end with digit 1 in factorial base].

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

A257692 Numbers such that the smallest nonzero digit present (A257679) in their factorial base representation is 2.

Original entry on oeis.org

4, 12, 16, 22, 48, 52, 60, 64, 66, 70, 76, 84, 88, 94, 100, 108, 112, 118, 240, 244, 252, 256, 258, 262, 288, 292, 300, 304, 306, 310, 312, 316, 324, 328, 330, 334, 336, 340, 348, 352, 354, 358, 364, 372, 376, 382, 408, 412, 420, 424, 426, 430, 436, 444, 448, 454, 460, 468, 472, 478, 484, 492, 496, 502

Offset: 1

Antti Karttunen, May 04 2015

Numbers k for which A257679(k) = 2.

			Factorial base representation (A007623) of 22 is "320" as 22 = 3*3! + 2*2! + 0*1!, thus a(22) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to factorial base representation.

Row 2 of A257503.
Cf. A007623, A257679, A257693.
Cf. also A257262.

q[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; !MemberQ[s, 1] && MemberQ[s, 2]]; Select[Range[500], q] (* Amiram Eldar, Feb 14 2024 *)

def A(n, p=2): return n if nIndranil Ghosh, Jun 19 2017

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Scheme

Formula

Extensions

A257679 The smallest nonzero digit present in the factorial base representation (A007623) of n, 0 if no nonzero digits present.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Scheme

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Scheme

Formula

Extensions

A257692 Numbers such that the smallest nonzero digit present (A257679) in their factorial base representation is 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python