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Base-n digit k involved in anomalous cancellation in the proper fraction A292288(n)/A292289(n).
3

%I #9 Sep 18 2017 18:13:36

%S 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,

%T 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,

%U 20,30,1,59,1,61,31,9,16,13,1,67,24,23,1,71,1,73,37

%N Base-n digit k involved in anomalous cancellation in the proper fraction A292288(n)/A292289(n).

%C 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.

%C 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

%H Michael De Vlieger, <a href="/A292393/b292393.txt">Table of n, a(n) for n = 2..120</a>

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/AnomalousCancellation.html">Anomalous Cancellation</a>

%F a(p) = 1.

%F a(p + 1) = p.

%e 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.

%e 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.

%e 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.

%e Table relating a(n) with A292288(n) and A292289(n).

%e n = base and index.

%e N = A292288(n) = smallest numerator that pertains to D.

%e D = A292289(n) = smallest denominator that has a nontrivial anomalous cancellation in base n.

%e n/d = simplified ratio of numerator N and denominator D.

%e k = a(n) = base-n digit anomalously canceled in the numerator and denominator to arrive at N/D.

%e .

%e n N D N/D k

%e ------------------------------

%e 2 3 6 1/2 1

%e 3 4 12 1/3 1

%e 4 7 14 1/2 3

%e 5 6 30 1/5 1

%e 6 11 33 1/3 5

%e 7 8 56 1/7 1

%e 8 15 60 1/4 7

%e 9 13 39 1/3 4

%e 10 16 64 1/4 6

%e 11 12 132 1/11 1

%e 12 23 138 1/6 11

%e 13 14 182 1/13 1

%e 14 27 189 1/7 13

%e 15 22 110 1/5 7

%e 16 21 84 1/4 5

%e 17 18 306 1/17 1

%e 18 35 315 1/9 17

%e 19 20 380 1/19 1

%e 20 39 390 1/10 19

%t 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 *)

%Y Cf. A292288 (numerators), A292289 (denominators).

%K nonn,base

%O 2,3

%A _Michael De Vlieger_, Sep 15 2017