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A292393
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Base-n digit k involved in anomalous cancellation in the proper fraction A292288(n)/A292289(n).
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3
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1, 1, 3, 1, 5, 1, 7, 4, 6, 1, 11, 1, 13, 7, 5, 1, 17, 1, 19, 8, 12, 1, 23, 6, 15, 13, 9, 1, 29, 1, 31, 12, 18, 17, 7, 1, 37, 19, 13, 1, 41, 1, 43, 11, 24, 1, 47, 8, 21, 20, 17, 1, 53, 12, 15, 20, 30, 1, 59, 1, 61, 31, 9, 16, 13, 1, 67, 24, 23, 1, 71, 1, 73, 37
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OFFSET
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2,3
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COMMENTS
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For prime base p, (p + 1)/(p^2 + p) simplifies to 1/p by canceling digit k = 1 in the numerator and denominator. This fraction is written "11/110" in base p and simplifies to "1/10" = 1/p.
See link "Base-b proper fractions n/d having nontrivial anomalous cancellation, with 2 <= b <= 120 and d <= b^2 + b" at A292289 for more information. - Michael De Vlieger, Sep 18 2017
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LINKS
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FORMULA
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a(p) = 1.
a(p + 1) = p.
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EXAMPLE
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a(10) = 6, since A292288(10)/A292289(10) = 16/64 = 1/4; we can "cancel" k = 6 in the numerator and the denominator and obtain 1/4 anomalously.
a(12) = 11, since A292288(12)/A292289(12) = 23/138 = "1b/b6" in base 12, where "b" represents digit 11. This fraction simplifies to 1/6. Digit "b" = 11 is canceled and "anomalously" yields 1/6.
a(16) = 5, since A292288(16)/A292289(16) = 21/84 = hexadecimal "15/54". This fraction simplifies to 1/4. We can "cancel" k = 5 in the numerator and denominator and obtain 1/4 anomalously.
n = base and index.
N = A292288(n) = smallest numerator that pertains to D.
D = A292289(n) = smallest denominator that has a nontrivial anomalous cancellation in base n.
n/d = simplified ratio of numerator N and denominator D.
k = a(n) = base-n digit anomalously canceled in the numerator and denominator to arrive at N/D.
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n N D N/D k
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2 3 6 1/2 1
3 4 12 1/3 1
4 7 14 1/2 3
5 6 30 1/5 1
6 11 33 1/3 5
7 8 56 1/7 1
8 15 60 1/4 7
9 13 39 1/3 4
10 16 64 1/4 6
11 12 132 1/11 1
12 23 138 1/6 11
13 14 182 1/13 1
14 27 189 1/7 13
15 22 110 1/5 7
16 21 84 1/4 5
17 18 306 1/17 1
18 35 315 1/9 17
19 20 380 1/19 1
20 39 390 1/10 19
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MATHEMATICA
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Table[Intersection[IntegerDigits[#1, b], IntegerDigits[#2, b]] & @@ Flatten@ Catch@ Do[If[Length@ # > 0, Throw[#], #] &@ 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^2 + b}], {b, 2, 30}] // Flatten (* Michael De Vlieger, Sep 15 2017 *)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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