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 Fractions With Nontrivial Anomalous Cancellation, Hey you can't do that! Wait, it works!
icarus
Posted: Sep 13 2017, 09:00 PM


Dozens Demigod


Group: Admin
Posts: 1,913
Member No.: 50
Joined: 11-April 06



I was working on a sequence Neil Sloane wrote (OEIS A291094). This is a sequence that contains denominators of fractions thus. Let n and d be the numerator and the denominator of a fraction respectively such that I can "cancel" a digit D in both n and in d and have the result = n/d. Usually this doesn't work for obvious reasons. 14/48, cancelling the 4s does not leave us with 7/24 but instead with 1/8, which is why we don't cancel digits when we are trying to simplify fractions! Famously, decimal 16/64, cancelling the 6s, actually does work: it does equal 1/4 like it is "supposed to". This is called "anomalous cancellation."

There is a trivial condition, that is the cancellation of trailing zeros. This is discounted regarding what we're interested in.

This sequence ports nicely into the dozenal condition and it might entertain folks here recreationally. The decimal case 16/64 = 1/4 is famous meme-wise; in dozenal the meme would be 1b;/b6; = 1/6.

The Wolfram language (11.1) code I wrote follows.

CODE

With[{b = 12},
 Apply[Join,
  Parallelize@
   Table[Map[{#, m} &, #] &@
     Select[Range[b + 1, m - 1],
      Function[k,
       Function[{r, w, n, d},
            AnyTrue[
             Flatten@
              Map[Apply[Outer[Divide, #1, #2] &, #] &,
               Transpose@
                MapAt[# /. 0 -> Nothing &,
                 Map[Function[x,
                   Map[Map[FromDigits[#, b] &@ Delete[x, #] &,
                   Position[x, #]] &, Intersection @@ {n, d}]], {n,
                   d}], -1]], # == Divide @@ {k, m} &]] @@ {k/m, #,
            First@ #, Last@ #} &@ Map[IntegerDigits[#, b] &, {k, m}] -
        Boole[Mod[{k, m}, b] == {0, 0}]] ], {m, b, b^3}]]]


Here are the first gross/biqua of fractions that enjoy nontrivial anomalous cancellation.

CODE

Dozenal fractions with nontrivial anomalous cancellation

All figures are dozenal after this sentence.

The trivial condition is that where N and D are both congruent to 0 (mod 10).

n = index.
N = Numerators of fractions with nontrivial anomalous cancellation.
D = Denominators of fractions with nontrivial anomalous cancellation.
   listed with multiplicity if multiple numerators are possible.
r = ratio N/D

 n     N     D   r
-----------------------
 1    1b    b6   1/6  
 2    2b    b8   1/4  
 3    3b    b9   1/3  
 4    5b    ba   1/2  
 5    11   110   1/10
 6    12   120   1/10
 7    22   121   2/11
 8    13   130   1/10
 9    33   132   3/12
 a    14   140   1/10
 b    44   143   4/13
10    15   150   1/10
11    55   154   5/14
12    16   160   1/10
13    66   165   6/15
14    17   170   1/10
15    77   176   7/16
16    18   180   1/10
17    88   187   8/17
18    19   190   1/10
19    99   198   9/18
1a    1a   1a0   1/10
1b    aa   1a9   a/19
20    1b   1b0   1/10
21    b6   1b0   1/2  
22    b7   1b2   1/2  
23    2b   1b4   1/8  
24    b8   1b4   1/2  
25    3b   1b6   1/6  
26    b9   1b6   1/2  
27    5b   1b8   1/4  
28    ba   1b8   1/2  
29    7b   1b9   1/3  
2a    bb   1ba   1/2  
2b   101   202   1/2  
30   102   204   1/2  
31   103   206   1/2  
32   104   208   1/2  
33   105   20a   1/2  
34    21   210   1/10
35    22   220   1/10
36   121   220   11/20
37    23   230   1/10
38    33   231   3/21
39   132   231   12/21
3a    24   240   1/10
3b    44   242   2/11
40   143   242   13/22
41    25   250   1/10
42    55   253   5/23
43   154   253   14/23
44    26   260   1/10
45    66   264   3/12
46   165   264   15/24
47    27   270   1/10
48    77   275   7/25
49   176   275   16/25
4a    28   280   1/10
4b    88   286   4/13
50   187   286   17/26
51    29   290   1/10
52    99   297   9/27
53   198   297   18/27
54    2a   2a0   1/10
55    aa   2a8   5/14
56   1a9   2a8   19/28
57    2b   2b0   1/10
58    b8   2b0   1/3  
59   1b4   2b0   2/3  
5a    3b   2b3   1/9  
5b    b9   2b3   1/3  
60   1b6   2b3   2/3  
61    5b   2b6   1/6  
62    ba   2b6   1/3  
63   1b8   2b6   2/3  
64    8b   2b8   1/4  
65    bb   2b9   1/3  
66   1ba   2b9   2/3  
67   15b   2ba   1/2  
68   101   303   1/3  
69   202   303   2/3  
6a   102   306   1/3  
6b   204   306   2/3  
70   103   309   1/3  
71   206   309   2/3  
72    31   310   1/10
73    32   320   1/10
74    33   330   1/10
75   132   330   7/16
76   231   330   21/30
77    34   340   1/10
78    44   341   4/31
79   143   341   13/31
7a   242   341   22/31
7b   139   346   7/16
80    35   350   1/10
81    55   352   5/32
82   154   352   8/17
83   253   352   23/32
84    36   360   1/10
85    66   363   2/11
86   165   363   15/33
87   264   363   24/33
88    37   370   1/10
89    77   374   7/34
8a   176   374   9/18
8b   275   374   25/34
90    38   380   1/10
91    88   385   8/35
92   187   385   17/35
93   286   385   26/35
94    39   390   1/10
95    99   396   3/12
96   198   396   a/19
97   297   396   27/36
98    3a   3a0   1/10
99    aa   3a7   a/37
9a   1a9   3a7   19/37
9b   2a8   3a7   28/37
a0    3b   3b0   1/10
a1    b9   3b0   1/4  
a2   1b6   3b0   1/2  
a3   2b3   3b0   3/4  
a4   1b7   3b2   1/2  
a5    5b   3b4   1/8  
a6    ba   3b4   1/4  
a7   1b8   3b4   1/2  
a8   2b6   3b4   3/4  
a9    7b   3b6   1/6  
aa   1b9   3b6   1/2  
ab    bb   3b8   1/4  
b0   1ba   3b8   1/2  
b1   2b9   3b8   3/4  
b2   13b   3b9   1/3  
b3   1bb   3ba   1/2  
b4   201   402   1/2  
b5   101   404   1/4  
b6   202   404   1/2  
b7   303   404   3/4  
b8   203   406   1/2  
b9   102   408   1/4  
ba   204   408   1/2  
bb   306   408   3/4  
100   205   40a   1/2  
Top
Double sharp
Posted: Sep 14 2017, 02:49 PM


Dozens Disciple


Group: Members
Posts: 1,402
Member No.: 1,150
Joined: 19-September 15



In the two-digit case, we are looking for solutions to (xb + y)/(zb + x) = y/z. As MathWorld notes, every proper divisor of the base b corresponds to one solution (and so prime bases have no solutions).

In the dozenal case, omega is prime, and so the divisor solutions (1b/b6 = 1/6, 2b/b8 = 1/4, 3b/b9 = 1/3, 5b/ba = 1/2) are the only solutions. In decimal, omega is composite, so while solutions do come this way (19/95 = 1/5, 49/95 = 1/2) they are not the only ones (16/64 = 1/4, 26/65 = 2/5). There are always an even number of solutions, unless the base is an even square (like 4, 16, or 36).
Top
icarus
Posted: Sep 14 2017, 03:29 PM


Dozens Demigod


Group: Admin
Posts: 1,913
Member No.: 50
Joined: 11-April 06



Precisely!

I have contributed the sequence that notes the smallest denominator d with an anomalous proper cancellation (OEIS A292289). Still have to finish the edit (b-file and chart). I will do the numerator this afternoon and have all cases for bases 2 <= b <= 120, generated overnight. Right now I am working on the number of denominators b < d <= b^2 + b (since primes p have smallest anomalous cancellation proper fraction "11/110" thus (p + 1)/(p^2 + p) = "1/10" = 1/p). From this I can sort how many two base-b digit proper anomalous fractions there are. I have a table similar to the one I am posting here that I will add to A292289 and A292288 once this data is done; it might take a day to generate despite parallelizing the function.

This is not the final document: the final will have 120 terms and the number of d < b^2 (two-digit proper fractions) and d <= b^2 + b.

CODE

Least numerator of the proper fraction having the smallest denominator
that has a nontrivial anomalous cancellation in base b >= 2.

A trivial anomalous cancellation involves digit k = 0 for numerator n and denominator d
both such that they are congruent to 0 (mod b).

b = base and index
n = A292288(b) = smallest numerator that pertains to d
d = A292289(b) = smallest denominator that has a nontrivial anomalous cancellation in base b
n/d = simplified ratio of numerator n and denominator d.
k = base-b digit cancelled in the numerator and denominator to arrive at n/d
b-n+1 = difference between base and numerator plus one.
b^2-d = difference between the square of the base and denominator.

 b     n       d   n/d       k   b-n+1   b^2-d
-----------------------------------------------
 2     3       6   1/2       1     0      -2
 3     4      12   1/3       1     0      -3
 4     7      14   1/2       3     2       2
 5     6      30   1/5       1     0      -5
 6    11      33   1/3       5     4       3
 7     8      56   1/7       1     0      -7
 8    15      60   1/4       7     6       4
 9    13      39   1/3       4     3      42
10    16      64   1/4       6     5      36
11    12     132   1/11      1     0     -11
12    23     138   1/6      11    10       6
13    14     182   1/13      1     0     -13
14    27     189   1/7      13    12       7
15    22     110   1/5       7     6     115
16    21      84   1/4       5     4     172
17    18     306   1/17      1     0     -17
18    35     315   1/9      17    16       9
19    20     380   1/19      1     0     -19
20    39     390   1/10     19    18      10
21    29     174   1/6       8     7     267
22    34     272   1/8      12    11     212
23    24     552   1/23      1     0     -23
24    47     564   1/12     23    22      12
25    31     155   1/5       6     5     470
26    67     402   1/6      15    40     274
27    40     360   1/9      13    12     369
28    37     259   1/7       9     8     525
29    30     870   1/29      1     0     -29
30    59     885   1/15     29    28      15
31    32     992   1/31      1     0     -31
32    63    1008   1/16     31    30      16
33    45     405   1/9      12    11     684
34    52     624   1/12     18    17     532
35    87     609   1/7      17    51     616
36    43     258   1/6       7     6    1038
37    38    1406   1/37      1     0     -37
38    75    1425   1/19     37    36      19
39    58     754   1/13     19    18     767
40    53     530   1/10     13    12    1070
41    42    1722   1/41      1     0     -41
42    83    1743   1/21     41    40      21
43    44    1892   1/43      1     0     -43
44    87    1914   1/22     43    42      22
45    56     504   1/9      11    10    1521
46    70    1120   1/16     24    23     996
47    48    2256   1/47      1     0     -47
48    95    2280   1/24     47    46      24
49    57     399   1/7       8     7    2002
50    71    1065   1/15     21    20    1435
51   122    1037   2/17     20    70    1564
52    69     897   1/13     17    16    1807
53    54    2862   1/53      1     0     -53
54   107    2889   1/27     53    52      27
55    67     670   1/10     12    11    2355
56    71     852   1/12     15    14    2284
57    77    1155   1/15     20    19    2094
58    88    1760   1/20     30    29    1604
59    60    3540   1/59      1     0     -59
60   119    3570   1/30     59    58      30

Observations:
1. For prime base p, (p + 1)/(p^2 + p) simplifies to 1/p by cancelling digit k = 1
  in the numerator and denominator.
2. Smallest base b for which n/d, simplified, has a numerator greater than 1 is 51.
  The next terms are 77 and 92.
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