# Fractions With Nontrivial Anomalous Cancellation

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
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: Select all

``````With&#91;{b = 12},
&nbsp;Apply&#91;Join,
&nbsp; Parallelize@
&nbsp; &nbsp;Table&#91;Map&#91;{#, m} &, #&#93; &@
&nbsp; &nbsp; &nbsp;Select&#91;Range&#91;b + 1, m - 1&#93;,
&nbsp; &nbsp; &nbsp; Function&#91;k,
&nbsp; &nbsp; &nbsp; &nbsp;Function&#91;{r, w, n, d},
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; AnyTrue&#91;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;Flatten@
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; Map&#91;Apply&#91;Outer&#91;Divide, #1, #2&#93; &, #&#93; &,
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;Transpose@
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; MapAt&#91;# /. 0 -> Nothing &,
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;Map&#91;Function&#91;x,
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;Map&#91;Map&#91;FromDigits&#91;#, b&#93; &@ Delete&#91;x, #&#93; &,
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;Position&#91;x, #&#93;&#93; &, Intersection @@ {n, d}&#93;&#93;, {n,
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;d}&#93;, -1&#93;&#93;, # == Divide @@ {k, m} &&#93;&#93; @@ {k/m, #,
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; First@ #, Last@ #} &@ Map&#91;IntegerDigits&#91;#, b&#93; &, {k, m}&#93; -
&nbsp; &nbsp; &nbsp; &nbsp; Boole&#91;Mod&#91;{k, m}, b&#93; == {0, 0}&#93;&#93; &#93;, {m, b, b^3}&#93;&#93;&#93;
``````
Here are the first gross/biqua of fractions that enjoy nontrivial anomalous cancellation.

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``````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 &#40;mod 10&#41;.

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

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

Dozens Demigod
Double sharp
Dozens Demigod
Joined: Sep 19 2015, 11:02 AM
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).

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
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.

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``````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 &#40;mod b&#41;.

b = base and index
n = A292288&#40;b&#41; = smallest numerator that pertains to d
d = A292289&#40;b&#41; = 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.

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

Observations&#58;
1. For prime base p, &#40;p + 1&#41;/&#40;p^2 + p&#41; simplifies to 1/p by cancelling digit k = 1
&nbsp; in the numerator and denominator.
2. Smallest base b for which n/d, simplified, has a numerator greater than 1 is 51.
&nbsp; The next terms are 77 and 92.
``````