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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).
+10
16
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
COMMENTS
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.
FORMULA
A(1,col) = A256450(col-1), and for row > 1, A(row,col) = A255411(A(row-1,col)).
EXAMPLE
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
...
PROG
(Scheme)
(define (A257503bi row col) (if (= 1 row) ( A256450 (- col 1)) ( A255411 (A257503bi (- row 1) col))))
CROSSREFS
Column 4: A213167 (without the initial one). Column 5: A052571 (without initial zeros).
The smallest nonzero digit present in the factorial base representation ( A007623) of n, 0 if no nonzero digits present.
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16
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
COMMENTS
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).
FORMULA
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].
EXAMPLE
Factorial base representation ( A007623) of 4 is "20", the smallest digit which is not zero is "2", thus a(4) = 2.
MATHEMATICA
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 *)
PROG
(Scheme)
(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:
(Python)
def A(n, p=2):
return n if n<p else A(n//p, p+1)*10 + n%p
def a(n):
return 0 if n==0 else min(int(i) for i in str(A(n)) if i !='0')
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.
+10
15
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
COMMENTS
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.
FORMULA
A(row,1) = A256450(row-1), and for col > 1, A(row,col) = A255411(A(row,col-1)).
EXAMPLE
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
...
PROG
(Scheme)
(define (A257505bi row col) (if (= 1 col) ( A256450 (- row 1)) ( A255411 (A257505bi row (- col 1)))))
CROSSREFS
Row 4: A213167 (without the initial one). Row 5: A052571 (without initial zeros).
Numbers such that the smallest nonzero digit present ( A257679) in their factorial base representation is 3.
+10
5
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
COMMENTS
Numbers k for which A257679(k) = 3.
EXAMPLE
Factorial base representation ( A007623) of 18 is "300" (as 18 = 3*3! + 0*2! + 0*1!), thus a(18) = 3.
MATHEMATICA
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 *)
PROG
(Python)
def A(n, p=2): return n if n<p else A(n//p, p+1)*10 + n%p
def a(n): return 0 if n==0 else min([int(i) for i in str(A(n)) if i !='0'])
print([n for n in range(1, 4001) if a(n)==3]) # Indranil Ghosh, Jun 19 2017
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