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%I #15 Sep 18 2017 17:27:07
%S 6,12,14,30,33,56,60,39,64,132,138,182,189,110,84,306,315,380,390,174,
%T 272,552,564,155,402,360,259,870,885,992,1008,405,624,609,258,1406,
%U 1425,754,530,1722,1743,1892,1914,504,1120,2256,2280,399,1065,1037,897,2862
%N Smallest denominator of a proper fraction that has a nontrivial anomalous cancellation in base b.
%C See comments at A291093.
%C 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.
%C Smallest base b for which n/d, simplified, has a numerator greater than 1 is 51.
%H Michael De Vlieger, <a href="/A292289/b292289.txt">Table of n, a(n) for n = 2..120</a>
%H Michael De Vlieger, <a href="/A292289/a292289.txt">Base-b proper fractions n/d having nontrivial anomalous cancellation, with 2 <= b <= 120 and d <= b^2 + b.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AnomalousCancellation.html">Anomalous Cancellation</a>
%F a(p) = p^2 + p.
%e 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.
%e Table of smallest values correlated with least numerators:
%e b = base and index.
%e n = smallest numerator that pertains to d.
%e d = smallest denominator that has a nontrivial anomalous cancellation in base b (this sequence).
%e n/d = simplified ratio of numerator n and denominator d.
%e k = base-b digit cancelled in the numerator and denominator to arrive at n/d.
%e b-n+1 = difference between base and numerator plus one.
%e b^2-d = difference between the square of the base and denominator.
%e .
%e b n d n/d k b-n+1 b^2-d
%e -----------------------------------------
%e 2 3 6 1/2 1 0 -2
%e 3 4 12 1/3 1 0 -3
%e 4 7 14 1/2 3 2 2
%e 5 6 30 1/5 1 0 -5
%e 6 11 33 1/3 5 4 3
%e 7 8 56 1/7 1 0 -7
%e 8 15 60 1/4 7 6 4
%e 9 13 39 1/3 4 3 42
%e 10 16 64 1/4 6 5 36
%e 11 12 132 1/11 1 0 -11
%e 12 23 138 1/6 11 10 6
%e 13 14 182 1/13 1 0 -13
%e 14 27 189 1/7 13 12 7
%e 15 22 110 1/5 7 6 115
%e 16 21 84 1/4 5 4 172
%t 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 *)
%Y Cf. A291093/A291094, A292288 (numerators), A292393 (digit that is canceled).
%K nonn,frac,base
%O 2,1
%A _Michael De Vlieger_, Sep 13 2017