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A292289
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Smallest denominator of a proper fraction that has a nontrivial anomalous cancellation in base b.
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3
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6, 12, 14, 30, 33, 56, 60, 39, 64, 132, 138, 182, 189, 110, 84, 306, 315, 380, 390, 174, 272, 552, 564, 155, 402, 360, 259, 870, 885, 992, 1008, 405, 624, 609, 258, 1406, 1425, 754, 530, 1722, 1743, 1892, 1914, 504, 1120, 2256, 2280, 399, 1065, 1037, 897, 2862
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OFFSET
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2,1
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COMMENTS
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For prime base p, (p + 1)/(p^2 + p) simplifies to 1/p by cancelling digit k = 1 in the numerator and denominator. This fraction is written "11/110" in base p and simplifies to "1/10" = 1/p.
Smallest base b for which n/d, simplified, has a numerator greater than 1 is 51.
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LINKS
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FORMULA
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a(p) = p^2 + p.
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EXAMPLE
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a(5) = 30, the corresponding numerator is 6; these are written "11/110" in quinary, cancelling a 1 in both numerator and denominator yields "1/10" which is 1/5. 6/30 = 1/5.
Table of smallest values correlated with least numerators:
b = base and index.
n = smallest numerator that pertains to d.
d = smallest denominator that has a nontrivial anomalous cancellation in base b (this sequence).
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.
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b n d n/d k b-n+1 b^2-d
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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
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MATHEMATICA
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Table[SelectFirst[Range[b, b^2 + b], Function[m, 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}]] ] != {}]], {b, 2, 30}] (* Michael De Vlieger, Sep 13 2017 *)
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CROSSREFS
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KEYWORD
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nonn,frac,base
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AUTHOR
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STATUS
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approved
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