A257679 -id:A257679 - cp's OEIS frontend

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

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

A257693 Numbers such that the smallest nonzero digit present (A257679) in their factorial base representation is 3.

Original entry on oeis.org

18, 72, 90, 114, 360, 378, 432, 450, 456, 474, 498, 552, 570, 594, 618, 672, 690, 714, 2160, 2178, 2232, 2250, 2256, 2274, 2520, 2538, 2592, 2610, 2616, 2634, 2640, 2658, 2712, 2730, 2736, 2754, 2760, 2778, 2832, 2850, 2856, 2874, 2898, 2952, 2970, 2994, 3240, 3258, 3312, 3330, 3336, 3354, 3378, 3432, 3450, 3474, 3498, 3552

Offset: 1

Antti Karttunen, May 04 2015

Numbers k for which A257679(k) = 3.

			Factorial base representation (A007623) of 18 is "300" (as 18 = 3*3! + 0*2! + 0*1!), thus a(18) = 3.

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

Row 3 of A257503.
Cf. A007623, A257679, A256450, A257692.
Cf. also A257263.

q[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; !ContainsAny[s, {1, 2}] && MemberQ[s, 3]]; Select[Range[3600], q] (* Amiram Eldar, Feb 14 2024 *)

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

A256450 Numbers that have at least one 1-digit in their factorial base representation (A007623).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 92, 93, 95, 97, 98, 99, 101

Offset: 0

Antti Karttunen, Apr 27 2015

Numbers n for which A257679(n) = 1, i.e., numbers n such that the least nonzero digit in their factorial base representation (A007623) is 1.
Involution A225901 maps each term of this sequence to a unique term of A273670, and vice versa.
Starting offset is zero (with a(0) = 1) because it is the most natural offset for the given fast recurrence.

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

Cf. A007623, A257679.
Complement of A255411.
Cf. A257680 (characteristic function), A273662 (left inverse).
First row of A257503, first column of A257505.
Subsequences: A059590 (apart from its zero-term), A255341, A255342, A255343, A257262, A257263, A258198, A258199.
Cf. also A227187 (numbers with at least one nonleading zero) and A273670, A225901.

Select[Range@ 101, MemberQ[IntegerDigits[#, MixedRadix[Reverse@ Range@ 12]], 1] &] (* Michael De Vlieger, May 30 2016, Version 10.2 *)
r = MixedRadix[Reverse@ Range[2, 12]]; Select[Range@ 101, Min[IntegerDigits[#, r] /. 0 -> Nothing] == 1 &]  (* Michael De Vlieger, Aug 14 2016, Version 10.2 *)

def A(n, p=2): return n if n=1]) # Indranil Ghosh, Jun 19 2017

a(0) = 1, and for n >= 1, if A257511(1+a(n-1)) > 0, then a(n) = a(n-1) + 1, otherwise a(n-1) + 2. [In particular, if the previous term is 2k, then the next term is 2k+1, because all odd numbers are members.]
Other identities:
For all n >= 0, A273662(a(n)) = n. [A273662 works as the left inverse for this sequence.]
From Antti Karttunen, May 26 2015: (Start)
Alternative recurrence for the same sequence:
Set k = A258198(n), d = n - A258199(n) and f = A000142(k+1) = (k+1)! If d < f then b(n) = f+d, otherwise b(n) = ((2+floor((d-f)/A258199(n))) * f) + b((d-f) mod A258199(n)). For offset=1 sequence, define a(n) = b(n-1).
(End)

Starting offset changed from 1 to 0 by Antti Karttunen, May 30 2016

A257684 Discard the rightmost digit from the factorial base representation of n and subtract one from all remaining nonzero digits, then convert back to decimal.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 13, 13, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 16, 16, 16, 16, 17, 17

Offset: 0

Antti Karttunen, May 04 2015

In other words, subtract one from all nonzero digits in the factorial base representation (A007623) of n and shift it one step right (i.e., delete the rightmost zero), then convert back to decimal.

			For 4, whose factorial base representation is "20" (as 4 = 2*2! + 0*1!), when we discard the rightmost zero, and subtract 1 from 2, we get "1", thus a(4) = 1.
For 18, whose factorial base representation is "300" (as 18 = 3*3! + 0*2! + 0*1!), when we discard the rightmost zero, and subtract 1 from 3, we get "20", thus a(18) = 4.

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

Positions of zeros: A059590.
Cf. A007623, A255411, A257685, A257687.
Can be used to define simple recurrences for sequences like A034968, A246359, A257679, A257694, A257695 and A257696.

nn = 95; m = 1; While[Factorial@ m < nn, m++]; m; Map[FromDigits[#, MixedRadix[Reverse@ Range[2, m]]] &[If[# == 0, 0, # - 1] & /@ Most@ IntegerDigits[#, MixedRadix[Reverse@ Range[2, m]]]] &, Range[0, nn]] (* Michael De Vlieger, Aug 11 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)))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 19 2017

(define (A257684 n) (let loop ((n n) (z 0) (i 2) (f 0)) (cond ((zero? n) z) (else (let ((d (remainder n i))) (loop (quotient n i) (+ z (* f (- d (if (zero? d) 0 1)))) (+ 1 i) (if (zero? f) 1 (* f (- i 1)))))))))

For all n >= 0, a(A255411(n)) = n. [This sequence works as a left inverse of A255411. See A257685 for a "cleaned up" version.]

A099563 a(0) = 0; for n > 0, a(n) = final nonzero number in the sequence n, f(n,2), f(f(n,2),3), f(f(f(n,2),3),4),..., where f(n,d) = floor(n/d); the most significant digit in the factorial base representation of n.

Original entry on oeis.org

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

Offset: 0

John W. Layman, Oct 22 2004

nonn

Records in {a(n)} occur at {1,4,18,96,600,4320,35280,322560,3265920,...}, which appears to be n*n! = A001563(n).

The most significant digit in the factorial expansion of n (A007623). Proof: The algorithm that computes the factorial expansion of n, generates the successive digits by repeatedly dividing the previous quotient with successively larger divisors (the remainders give the digits), starting from n itself and divisor 2. As a corollary we find that A001563 indeed gives the positions of the records. - Antti Karttunen, Jan 01 2007.

			For n=15, f(15,2) = floor(15/2)=7, f(7,3)=2, f(2,4)=0, so a(15)=2.
From _Antti Karttunen_, Dec 24 2015: (Start)
Example illustrating the role of this sequence in factorial base representation:
   n  A007623(n)       a(n) [= the most significant digit].
   0 =   0               0
   1 =   1               1
   2 =  10               1
   3 =  11               1
   4 =  20               2
   5 =  21               2
   6 = 100               1
   7 = 101               1
   8 = 110               1
   9 = 111               1
  10 = 120               1
  11 = 121               1
  12 = 200               2
  13 = 201               2
  14 = 210               2
  15 = 211               2
  16 = 220               2
  17 = 221               2
  18 = 300               3
  etc.
Note that there is no any upper bound for the size of digits in this representation.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..10080
Index entries for sequences related to factorial base representation

Cf. A001563, A007623, A099564.

Cf. also A034968, A048764, A051683, A055881, A126307, A230420, A246359, A249069, A257679, A257684, A257686, A257687, A265890, A265891, A265894, A265333, A265334.

Table[Floor[n/#] &@ (k = 1; While[(k + 1)! <= n, k++]; k!), {n, 0, 120}] (* Michael De Vlieger, Aug 30 2016 *)

A099563(n) = { my(i=2,dig=0); until(0==n, dig = n % i; n = (n - dig)/i; i++); return(dig); }; \\ Antti Karttunen, Dec 24 2015

def a(n):
    i=2
    d=0
    while n:
        d=n%i
        n=(n - d)//i
        i+=1
    return d
print([a(n) for n in range(201)]) # Indranil Ghosh, Jun 21 2017, after PARI code

(define (A099563 n) (let loop ((n n) (i 2)) (let* ((dig (modulo n i)) (next-n (/ (- n dig) i))) (if (zero? next-n) dig (loop next-n (+ 1 i))))))
(definec (A099563 n) (cond ((zero? n) n) ((= 1 (A265333 n)) 1) (else (+ 1 (A099563 (A257684 n)))))) ;; Based on given recurrence, using the memoization-macro definec
;; Antti Karttunen, Dec 24-25 2015

From Antti Karttunen, Dec 25 2015: (Start)
a(0) = 0; for n >= 1, if A265333(n) = 1 [when n is one of the terms of A265334], a(n) = 1, otherwise 1 + a(A257684(n)).
Other identities. For all n >= 0:
a(A001563(n)) = n. [Sequence works as a left inverse for A001563.]
a(n) = A257686(n) / A048764(n).
(End)

a(0) = 0 prepended and the alternative description added to the name-field by Antti Karttunen, Dec 24 2015

A257687 Discard the most significant digit from factorial base representation of n, then convert back to decimal: a(n) = n - A257686(n).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 04 2015

A060130(n) gives the number of steps needed to reach zero, when starting iterating as a(k), a(a(k)), etc., from the starting value k = n.

			Factorial base representation (A007623) of 1 is "1", discarding the most significant digit leaves nothing, taken to be zero, thus a(1) = 0.
Factorial base representation of 2 is "10", discarding the most significant digit leaves "0", thus a(2) = 0.
Factorial base representation of 3 is "11", discarding the most significant digit leaves "1", thus a(3) = 1.
Factorial base representation of 4 is "20", discarding the most significant digit leaves "0", thus a(4) = 0.

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

Cf. A007623, A257686.
Can be used (together with A099563) to define simple recurrences for sequences like A034968, A060130, A227153, A246359, A257511, A257679, A257680.
Cf. also A257684.

f[n_] := Block[{m = p = 1}, While[p*(m + 1) <= n, p = p*m; m++]; Mod[n, p]]; Array[f, 101, 0] (* Robert G. Wilson v, Jul 21 2015 *)

from sympy import factorial as f
def a007623(n, p=2): return n if nIndranil Ghosh, Jun 21 2017

Scheme
```
(define (A257687 n) (- n (A257686 n)))
```

a(n) = n - A257686(n).

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).

Original entry on oeis.org

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

A278236 Filter-sequence for factorial base (digit values): least number with the same prime signature as A276076(n).

Original entry on oeis.org

1, 2, 2, 6, 4, 12, 2, 6, 6, 30, 12, 60, 4, 12, 12, 60, 36, 180, 8, 24, 24, 120, 72, 360, 2, 6, 6, 30, 12, 60, 6, 30, 30, 210, 60, 420, 12, 60, 60, 420, 180, 1260, 24, 120, 120, 840, 360, 2520, 4, 12, 12, 60, 36, 180, 12, 60, 60, 420, 180, 1260, 36, 180, 180, 1260, 900, 6300, 72, 360, 360, 2520, 1800, 12600, 8, 24, 24, 120, 72, 360, 24, 120, 120, 840, 360, 2520

Offset: 0

Antti Karttunen, Nov 16 2016

This sequence can be used for filtering certain factorial base related sequences, because it matches only with any such sequence b that can be computed as b(n) = f(A276076(n)), where f(n) is any function that depends only on the prime signature of n (some of these are listed under the index entry for "sequences computed from exponents in ...").
Matching in this context means that the sequence a matches with the sequence b iff for all i, j: a(i) = a(j) => b(i) = b(j). In other words, iff the sequence b partitions the natural numbers to the same or coarser equivalence classes (as/than the sequence a) by the distinct values it obtains.
Any such sequence should match where the result is computed from the nonzero digits (that may also be > 9) in the factorial base representation of n, but does not depend on their order. Some of these are listed on the last line of the Crossrefs section.
Note that as A275735 is present in that list it means that the sequences matching to its filter-sequence A278235 form a subset of the sequences matching to this sequence. Also, for A275735 there is a stronger condition that for any i, j: a(i) = a(j) <=> A275735(i) = A275735(j), which if true, would imply that there is an injective function f such that f(A275735(n)) = A278236(n), and indeed, this function seems to be A181821.

Cf. A007623, A046523, A181821, A276076.
Similar sequences: A278222 (base-2 related), A069877 (base-10), A278226 (primorial base), A278225, A278234, A278235 (other variants for factorial base),
Differs from A278226 for the first time at n=24, where a(24)=2, while A278226(24)=16.
Sequences that partition N into same or coarser equivalence classes: A275735 (<=>), A034968, A060130, A227153, A227154, A246359, A257079, A257511, A257679, A257694, A257695, A257696, A264990, A275729, A275806, A275948, A275964, A278235.

a[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; s = ReverseSort[s]; Times @@ (Prime[Range[Length[s]]] ^ s)]; Array[a, 100, 0] (* Amiram Eldar, Feb 07 2024 *)

(define (A278236 n) (A046523 (A276076 n)))

a(n) = A046523(A276076(n)).

a(n) = A181821(A275735(n)). [Empirical formula found with the help of equivalence class matching. Not yet proved.]

A257681 Column index of A257503: If A257685(n) = 0, then a(n) = A257682(n), otherwise a(n) = a(A257685(n)).

Original entry on oeis.org

0, 1, 2, 3, 1, 4, 5, 6, 7, 8, 9, 10, 2, 11, 12, 13, 3, 14, 1, 15, 16, 17, 4, 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, 5, 43, 44, 45, 6, 46, 47, 48, 49, 50, 51, 52, 7, 53, 54, 55, 8, 56, 9, 57, 58, 59, 10, 60, 2

Offset: 0

Antti Karttunen, May 04 2015

nonn

a(0) = 0 by convention, because 0 is not present in arrays A257503 and A257505.

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

Cf. A257680, A257682, A257685.
Column index of A257503, row index of A257505.
Cf. also A257679 (gives the other index).

If A257685(n) = 0, then a(n) = A257682(n), otherwise a(n) = a(A257685(n)).

cp's OEIS Frontend

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

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A257693 Numbers such that the smallest nonzero digit present (A257679) in their factorial base representation is 3.

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A256450 Numbers that have at least one 1-digit in their factorial base representation (A007623).

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A257684 Discard the rightmost digit from the factorial base representation of n and subtract one from all remaining nonzero digits, then convert back to decimal.

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A099563 a(0) = 0; for n > 0, a(n) = final nonzero number in the sequence n, f(n,2), f(f(n,2),3), f(f(f(n,2),3),4),..., where f(n,d) = floor(n/d); the most significant digit in the factorial base representation of n.

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A257687 Discard the most significant digit from factorial base representation of n, then convert back to decimal: a(n) = n - A257686(n).

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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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